Unit 8 test: Applications of Integration

Find the average value of $f(x) = 3x^2$ on the interval $[0, 2]$.

A continuous function $g$ satisfies $\displaystyle\int_1^5 g(x)\,dx = 12$. Which statement must be true?

A particle moves along the $x$-axis with velocity $v(t) = t^2 - 6t + 8$ for $0 \le t \le 5$. Find the total distance the particle travels. Give an exact answer.

A tank holds 200 gallons of water at $t = 0$. Water flows in at a constant 30 gallons per minute and drains at $D(t) = 2t + 10$ gallons per minute. How many gallons are in the tank at $t = 8$ minutes?

A particle moves along a line with velocity $v(t)$ and position $x(t)$, where $x(2) = 7$. Which expression gives $x(6)$?

Find the area of the region bounded by $y = x^2$ and $y = x + 6$.

Which expression gives the area between $y = \sin x$ and $y = \cos x$ on $[0, \pi]$?

Find the area of the region bounded by $x = y^2$ and $x = 3y$.

The base of a solid is the region bounded by $y = \sqrt{x}$, the $x$-axis and $x = 4$. Cross sections perpendicular to the $x$-axis are semicircles. Find the exact volume.

The base of a solid is the region between $y = x^2$ and $y = 1$. Cross sections perpendicular to the $x$-axis are equilateral triangles. Find the exact volume.

The region under $y = x^2$ for $0 \le x \le 1$ is revolved around the $x$-axis. Find the exact volume.

The region in the first quadrant bounded by $y = \ln x$, the $x$-axis, the $y$-axis and the line $y = 2$ is revolved around the $y$-axis. Which integral gives the volume?

The region in the first quadrant between $y = x$ and $y = x^3$ is revolved around the $x$-axis. Find the exact volume.

The region between $y = x^2$ and $y = 1$ is revolved around the line $y = 3$. Find the exact volume.