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SL 5.5—Introduction to integration - IB Questionbank | RevisionDojo

Practice SL 5.5—Introduction to integration with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

The graphs of $y = 6 - x$ and $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

Calculate the area of the shaded region in Diagram 1.

[3]

Write down an integral for the area of the shaded region in Diagram 2.

[1]

Calculate the area of the shaded region in Diagram 2.

[3]

Hence, determine the area enclosed between the line $y = 6 - x$ and the curve $y = 1.5x^{2} - 2.5x + 3$ .

[1]

Question 2

HLPaper 2

A particle $P$ moves along the $x$ -axis. The velocity of $P$ is $v \, \text{m s}^{-1}$ at time $t$ seconds, where $v = -2t^2 + 16t - 24$ for $t \geq 0$ .

Find the times when $P$ is at instantaneous rest.

[2]

Find the magnitude of the particle's acceleration at 6 seconds.

[3]

Find the greatest speed of $P$ in the interval $0 \leq t \leq 6$ .

[2]

The particle starts from the origin $O$ . Find an expression for the displacement of $P$ from $O$ at time $t$ seconds.

[4]

Find the total distance travelled by $P$ in the interval $0 \leq t \leq 4$ .

[3]

Question 3

SLPaper 2

The function $g(x) = 3x^2 - 4x + 2$ models the rate of growth of a plant species over time, where $x$ is measured in weeks.

Calculate the indefinite integral of $g(x)$ .

[3]

Determine the definite integral of $g(x)$ from $x = 1$ to $x = 3$ .

[4]

Interpret the result of the definite integral in the context of the plant's growth.

[3]

Determine if $g(x)$ is increasing or decreasing on the interval [1, 3].

[4]

Verify your definite integral result using technology.

[4]

Question 4

SLPaper 1

Find $\int x \mathrm{e}^{x^2 - 1} \mathrm{d}x$ .

[4]

Find $f(x)$ , given that $f'(x) = x \mathrm{e}^{x^2 - 1}$ and $f(-1) = 3$ .

[3]

Question 5

HLPaper 1

Let $g'(x)=\frac{3x^2}{(x^3+1)^5}$ . Given that $g(0)=1$ , find $g(x)$ .

[3]

Question 6

SLPaper 1

Let $f'(x) = \sin^3(2x)\cos(2x)$ . Find $f(x)$ , given that $f\left(\frac{\pi}{4}\right) = 1$ .

[5]

Question 7

SLPaper 2

A fitness app calculates the energy expenditure of a user based on their running distance. The energy expenditure can be modeled by the integral $\int \frac{2 x}{\sqrt{1-x^2}} d x$ , where $x$ represents a normalized distance covered by the user, constrained within the interval $-1 < x < 1$ .

Find the indefinite integral of the energy expenditure function.

[4]

Calculate the definite integral of the energy expenditure function from $x=0$ to $x=\frac{1}{2}$ .

[3]

Question 8

SLPaper 2

A sector of a circle, centre $O$ and radius $4.5 \text{ m}$ , is shown in the following diagram.

A square field with side $8 \text{ m}$ has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to $4.5 \text{ m}$ from the post.

Let $V$ be the volume of grass eaten by the goat, in cubic metres, and $t$ be the length of time, in hours, that the goat has been in the field. The goat eats grass at the rate of $\frac{dV}{dt} = 0.3te^{-t}$ .

Find the angle $A\hat{O}B$ .

[3]

Find the area of the shaded segment.

[6]

Find the area of a circle with radius $4.5 \text{ m}$ .

[2]

Find the area of the field that can be reached by the goat.

[3]

Find the value of $t$ at which the goat is eating grass at the greatest rate.

[2]

Question 9

SLPaper 2

The function $f(x) = 3x^2 - 6x + 4$ models the growth rate of a plant species in a greenhouse.

Calculate the indefinite integral of $f(x)$ .

[2]

Using the boundary condition that $F(1) = 10$ , determine the constant of integration.

[2]

Find the definite integral of $f(x)$ from $x = 0$ to $x = 3$ to determine the total growth over this period.

[2]

Question 10

SLPaper 1

A researcher is investigating the rate of decay of a certain radioactive substance, which can be modeled by the function $h(x) = \frac{1}{\sqrt{x}}$ for $x > 0$ , where $x$ represents time in hours.

Find the indefinite integral of $h(x)$ .

[2]

Determine the area under the curve of $h(x)$ from $x = 1$ to $x = 4$ .

[2]

Paper

Level

Question 1

SLPaper 1

The graphs of $y = 6 - x$ and $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

Calculate the area of the shaded region in Diagram 1.

[3]

Write down an integral for the area of the shaded region in Diagram 2.

[1]

Calculate the area of the shaded region in Diagram 2.

[3]

Hence, determine the area enclosed between the line $y = 6 - x$ and the curve $y = 1.5x^{2} - 2.5x + 3$ .

[1]

Question 2

HLPaper 2

A particle $P$ moves along the $x$ -axis. The velocity of $P$ is $v \, \text{m s}^{-1}$ at time $t$ seconds, where $v = -2t^2 + 16t - 24$ for $t \geq 0$ .

Find the times when $P$ is at instantaneous rest.

[2]

Find the magnitude of the particle's acceleration at 6 seconds.

[3]

Find the greatest speed of $P$ in the interval $0 \leq t \leq 6$ .

[2]

The particle starts from the origin $O$ . Find an expression for the displacement of $P$ from $O$ at time $t$ seconds.

[4]

Find the total distance travelled by $P$ in the interval $0 \leq t \leq 4$ .

[3]

Question 3

SLPaper 2

The function $g(x) = 3x^2 - 4x + 2$ models the rate of growth of a plant species over time, where $x$ is measured in weeks.

Calculate the indefinite integral of $g(x)$ .

[3]

Determine the definite integral of $g(x)$ from $x = 1$ to $x = 3$ .

[4]

Interpret the result of the definite integral in the context of the plant's growth.

[3]

Determine if $g(x)$ is increasing or decreasing on the interval [1, 3].

[4]

Verify your definite integral result using technology.

[4]

Question 4

SLPaper 1

Find $\int x \mathrm{e}^{x^2 - 1} \mathrm{d}x$ .

[4]

Find $f(x)$ , given that $f'(x) = x \mathrm{e}^{x^2 - 1}$ and $f(-1) = 3$ .

[3]

Question 5

HLPaper 1

Let $g'(x)=\frac{3x^2}{(x^3+1)^5}$ . Given that $g(0)=1$ , find $g(x)$ .

[3]

Question 6

SLPaper 1

Let $f'(x) = \sin^3(2x)\cos(2x)$ . Find $f(x)$ , given that $f\left(\frac{\pi}{4}\right) = 1$ .

[5]

Question 7

SLPaper 2

Find the indefinite integral of the energy expenditure function.

[4]

Calculate the definite integral of the energy expenditure function from $x=0$ to $x=\frac{1}{2}$ .

[3]

Question 8

SLPaper 2

A sector of a circle, centre $O$ and radius $4.5 \text{ m}$ , is shown in the following diagram.

A square field with side $8 \text{ m}$ has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to $4.5 \text{ m}$ from the post.

Find the angle $A\hat{O}B$ .

[3]

Find the area of the shaded segment.

[6]

Find the area of a circle with radius $4.5 \text{ m}$ .

[2]

Find the area of the field that can be reached by the goat.

[3]

Find the value of $t$ at which the goat is eating grass at the greatest rate.

[2]

Question 9

SLPaper 2

The function $f(x) = 3x^2 - 6x + 4$ models the growth rate of a plant species in a greenhouse.

Calculate the indefinite integral of $f(x)$ .

[2]

Using the boundary condition that $F(1) = 10$ , determine the constant of integration.

[2]

Find the definite integral of $f(x)$ from $x = 0$ to $x = 3$ to determine the total growth over this period.

[2]

Question 10

SLPaper 1

A researcher is investigating the rate of decay of a certain radioactive substance, which can be modeled by the function $h(x) = \frac{1}{\sqrt{x}}$ for $x > 0$ , where $x$ represents time in hours.

Find the indefinite integral of $h(x)$ .

[2]

Determine the area under the curve of $h(x)$ from $x = 1$ to $x = 4$ .

[2]

Paper

Level

Question 1

SLPaper 1

The graphs of $y = 6 - x$ and $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

Calculate the area of the shaded region in Diagram 1.

[3]

Write down an integral for the area of the shaded region in Diagram 2.

[1]

Calculate the area of the shaded region in Diagram 2.

[3]

Hence, determine the area enclosed between the line $y = 6 - x$ and the curve $y = 1.5x^{2} - 2.5x + 3$ .

[1]

Question 2

HLPaper 2

A particle $P$ moves along the $x$ -axis. The velocity of $P$ is $v \, \text{m s}^{-1}$ at time $t$ seconds, where $v = -2t^2 + 16t - 24$ for $t \geq 0$ .

Find the times when $P$ is at instantaneous rest.

[2]

Find the magnitude of the particle's acceleration at 6 seconds.

[3]

Find the greatest speed of $P$ in the interval $0 \leq t \leq 6$ .

[2]

The particle starts from the origin $O$ . Find an expression for the displacement of $P$ from $O$ at time $t$ seconds.

[4]

Find the total distance travelled by $P$ in the interval $0 \leq t \leq 4$ .

[3]

Question 3

SLPaper 2

The function $g(x) = 3x^2 - 4x + 2$ models the rate of growth of a plant species over time, where $x$ is measured in weeks.

Calculate the indefinite integral of $g(x)$ .

[3]

Determine the definite integral of $g(x)$ from $x = 1$ to $x = 3$ .

[4]

Interpret the result of the definite integral in the context of the plant's growth.

[3]

Determine if $g(x)$ is increasing or decreasing on the interval [1, 3].

[4]

Verify your definite integral result using technology.

[4]

Question 4

SLPaper 1

Find $\int x \mathrm{e}^{x^2 - 1} \mathrm{d}x$ .

[4]

Find $f(x)$ , given that $f'(x) = x \mathrm{e}^{x^2 - 1}$ and $f(-1) = 3$ .

[3]

Question 5

HLPaper 1

Let $g'(x)=\frac{3x^2}{(x^3+1)^5}$ . Given that $g(0)=1$ , find $g(x)$ .

[3]

Question 6

SLPaper 1

Let $f'(x) = \sin^3(2x)\cos(2x)$ . Find $f(x)$ , given that $f\left(\frac{\pi}{4}\right) = 1$ .

[5]

Question 7

SLPaper 2

Find the indefinite integral of the energy expenditure function.

[4]

Calculate the definite integral of the energy expenditure function from $x=0$ to $x=\frac{1}{2}$ .

[3]

Question 8

SLPaper 2

A sector of a circle, centre $O$ and radius $4.5 \text{ m}$ , is shown in the following diagram.

A square field with side $8 \text{ m}$ has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to $4.5 \text{ m}$ from the post.

Find the angle $A\hat{O}B$ .

[3]

Find the area of the shaded segment.

[6]

Find the area of a circle with radius $4.5 \text{ m}$ .

[2]

Find the area of the field that can be reached by the goat.

[3]

Find the value of $t$ at which the goat is eating grass at the greatest rate.

[2]

Question 9

SLPaper 2

The function $f(x) = 3x^2 - 6x + 4$ models the growth rate of a plant species in a greenhouse.

Calculate the indefinite integral of $f(x)$ .

[2]

Using the boundary condition that $F(1) = 10$ , determine the constant of integration.

[2]

Find the definite integral of $f(x)$ from $x = 0$ to $x = 3$ to determine the total growth over this period.

[2]

Question 10

SLPaper 1

A researcher is investigating the rate of decay of a certain radioactive substance, which can be modeled by the function $h(x) = \frac{1}{\sqrt{x}}$ for $x > 0$ , where $x$ represents time in hours.

Find the indefinite integral of $h(x)$ .

[2]

Determine the area under the curve of $h(x)$ from $x = 1$ to $x = 4$ .

[2]

SL 5.5—Introduction to integration Questionbank