IB Mathematics Applications & Interpretation Question Type 1: Estimating Area with Equal Width Intervals Using Trapezoids Exercises

The following table shows the values of a function $f(x)$ for $0\le x\le 4$.

$$\begin{array}{cccccc}x& 0& 1& 2& 3& 4\\ f(x)& 2& 4& 7& 11& 16\end{array}$$

Approximate the area under the curve $y=f(x)$ from $x=0$ to $x=4$ using the trapezoidal rule.

Approximate the integral ${\int }_0^4(5e^{-x}+e^x)dx$ using the trapezoidal rule with 8 equal subintervals.

Approximate ${\int }_0^1{\text{e}}^x\text{d}x$ using the trapezoidal rule with 4 equal subintervals, given the following data:

$$\begin{array}{cccccc}x& 0& 0.25& 0.5& 0.75& 1\\ f(x)={\text{e}}^x& 1& 1.2840& 1.6487& 2.1170& 2.7183\end{array}$$

Use the trapezoidal rule with 4 subintervals to approximate ${\int }_0^{\pi }\sin xdx$.

The values of $\sin x$ are given in the table below at intervals of $0.5$ for $0\le x\le 2$.

Use the trapezoidal rule to find an approximate value for ${\int }_0^2\sin x\mathrm{d}x$.

Use the trapezoidal rule with 3 subintervals to approximate ${\int }_0^9\sqrt{x}dx$.

Approximate the integral ${\int }_0^4(5e^{-x}+e^x)dx$ using the trapezoidal rule with 4 equal subintervals.

Use the trapezoidal rule with 3 equal subintervals to approximate the integral ${\int }_1^{4}f(x)dx$ using the data provided in the table below:

$$\begin{array}{ccccc}x& 1& 2& 3& 4\\ f(x)& 10& 20& 25& 22\end{array}$$

Approximate the integral ${\int }_0^4(5e^{-x}+e^x)dx$ using the trapezoidal rule with 2 equal subintervals.

Approximate ${\int }_1^5\frac{1}{x}\text{d}x$ using the trapezoidal rule with 4 equal subintervals.

The speed of a car $v$, in $\text{m}{\text{s}}^{-1}$, is recorded at $5$-minute intervals for $20$ minutes. The data is shown in the following table.

Use the trapezoidal rule to estimate the total distance traveled by the car during the $20$-minute period. Give your answer in metres ($\text{m}$).