Question Type 3: Calculating the area between a polynomial in the positive section and the x axis Exercises

Find the area under the curve $y = 5x^4 - 2x^2 + 3$ from $x = -0.5$ to $x = 1.5$.

Find the exact area under $y = e^{x}\cos x$ from $x = 0$ to $x = \tfrac{\pi}{4}$.

Determine the area under $y = \ln(x+1) + x^2$ from $x = 0$ to $x = 1$.

Compute the area under $y = 4 - x^2$ from $x = -1$ to $x = 1$.

Calculate the area under the curve $y = 2x^2 + 3x + 2$ between $x = 0$ and $x = 2$.

Calculate the area between the curve $y = x^5 - x^3 + 2$ and the $x$-axis for $1 \leq x \leq 2$.

Use technology to evaluate the definite integral $\int_{0}^{1} (e^x \ln(2x) - x^4) \, dx$. Give your answer correct to three decimal places.

Find the area bounded by the curve $y = x^3 - 6x^2 + 9x + 1$ and the $x$-axis over the interval $[0,3]$.

Calculate the area under $y = e^{-x}\sin x$ from $x = 0$ to $x = \tfrac{\pi}{2}$.

Determine the area under the curve $y = 3x^3 + x$ on $[0,2]$.

Compute the area under the curve $y = x e^{x}$ between $x = 0$ and $x = 1$.

Find the exact area under $y = e^{2x} - x^3$ from $0$ to $1$.