Unit 4 test: Applications of Integration

Find the area of the region bounded by $y = x^2$ and $y = 8 - x^2$.

The region $R$ is bounded by $x = y^2 - 3$ and $x = y - 1$. Which integral gives the area of $R$?

Find the area of the region bounded by $y = e^x$, the line $y = e$, and the $y$-axis.

Calculator allowed. Find the area of the region enclosed by $y = \cos x$ and $y = x^2$. Round to three decimal places.

The base of a solid is the disk $x^2 + y^2 \le 4$. Cross sections perpendicular to the $x$-axis are squares. Find the volume.

The region under $y = \dfrac{1}{x}$ from $x = 1$ to $x = 3$ is revolved about the $x$-axis. Find the exact volume.

The region between $y = \sqrt{x}$ and $y = x$ is revolved about the $x$-axis. Find the exact volume.

The region bounded by $y = x^2$ and $y = 1$ is revolved about the line $y = 2$. What is the volume?

The region bounded by $y = x^3$, the line $y = 8$ and the $y$-axis is revolved about the $y$-axis. Find the exact volume.

Which integral gives the length of the curve $y = e^{2x}$ from $x = 0$ to $x = 1$?

Find the length of $y = \tfrac{1}{3}(x^2 + 2)^{3/2}$ from $x = 0$ to $x = 3$.

Find the exact length of the curve $x = \dfrac{y^2}{2} - \dfrac{\ln y}{4}$ from $y = 1$ to $y = e$.

Calculator allowed. Find the length of $y = \dfrac{1}{x}$ from $x = 1$ to $x = 3$. Round to three decimal places.