Question Type 2: Calculating the area between a function and the x-axis Exercises

Calculate the exact value of the definite integral $\displaystyle\int_{1}^{4}(x^2 - 5x)\,dx$ and state its geometric interpretation.

Find the area enclosed by the curve $y = x^2 - 5x$, the lines $x=2.5$, $x=7.5$, and the x-axis.

Find the area bounded by the y-axis, the curve $y = e^{x/2}$, and the vertical line $x=4$.

Calculate the area enclosed by the x-axis and the curve $y = x^2 - 5x$ between its x-intercepts.

Set up and evaluate the area enclosed by the y-axis and the curve $y = e^{x/2}$ for $y$ from $1$ to $e$.

Calculate the net signed area between the x-axis and the curve $y = x^2 - 5x$ from $x=0$ to $x=7$.

Calculate the total (unsigned) area between the x-axis and the curve $y = x^2 - 5x$ on the interval $[-1,6]$.

Calculate the area of the region bounded by the curve $y=e^{x/2}$, the y-axis, and the horizontal lines $y=2$ and $y=e^2$.

A region is bounded by the curve $y=e^{x/2}$, the x-axis, the y-axis, and the line $x=2$. Find its exact area.

Find the area enclosed by the curve $y=e^{x/2}$, the y-axis, and the line $y=4$.

Find the area of the region enclosed by the lines $y=4$, $y=e^{x/2}$, $x=0$, and $x=1$.