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

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

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 $y$-axis and the curve $y = \text{e}^{x/2}$ for $1 \le y \le \text{e}$.

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

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.

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

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 curve $y = e^{x/2}$ and the lines $y = 4$, $x = 0$, and $x = 1$.

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

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