Unit 8 check

Unit 8 check: Applications of Integration

12 questions from 12 lessons in Unit 8. Answer each one, and if you miss, follow the link back to where it is taught, then try again. Your score counts first tries.

Question 1 · from 8.1 · Calculate

No calculator. What is the average value of $f (x) = x^{2} + 2$ on the interval $[1, 4]$ ?

Your answer

Question 2 · from 8.2 · Calculate

No calculator. A particle moves along a line with velocity $v (t) = 3 t^{2} - 6 t$ for $0 \leq t \leq 3$ . What is the total distance the particle travels from $t = 0$ to $t = 3$ ?

Your answer

Question 3 · from 8.3 · Sort it

No calculator. Rain falls into a pond at a rate of $F (t)$ gallons per hour, water drains out at $D (t)$ gallons per hour, and the pond holds 9,000 gallons at $t = 0$ . Tap a question, then the work it needs.

Evaluate a rate at one time

Integrate a rate over an interval

Add a starting amount to an integral

Question 4 · from 8.4 · Sort it

No calculator. The region $R$ is bounded by the graphs of $y = 4 - x^{2}$ and $y = x + 2$ , which meet at $(- 2, 0)$ and $(1, 3)$ . Sort each expression: does it give the area of $R$ , or not? Tap a card, then a bin.

Gives the area of $R$

Does not give the area of $R$

Question 5 · from 8.5 · Decide and justify

No calculator. Let $R$ be the region bounded by $y = x^{2}$ and $y = 2 x$ . Its area is $\int_{0}^{2} (2 x - x^{2}) d x$ . A classmate says the same area is $\int_{0}^{4} (\sqrt{y} - \frac{y}{2}) d y$ . Is the classmate right?

Which reason supports your answer?

Setup for question 6

The functions $f$ and $g$ are continuous on $[0, 6]$ , and their graphs meet only at $x = 1$ and $x = 4$ on that interval. The table gives the definite integral of $f (x) - g (x)$ over each piece.

The integral of f minus g over each piece
Interval	$\int (f (x) - g (x)) d x$ over the interval
$0 \leq x \leq 1$	2.5
$1 \leq x \leq 4$	$- 6$
$4 \leq x \leq 6$	1.5

Question 6 · from 8.6 · Calculate

No calculator. What is the area of the region between the graphs of $f$ and $g$ from $x = 0$ to $x = 6$ ?

Your answer

Setup for question 7

A woodworker shapes a square table leg 40 centimeters long. At a distance of $x$ centimeters from the top of the leg, its cross section perpendicular to the leg's length is a square with side $s (x)$ centimeters, where $s$ is continuous. Some values are in the table.

Side of the leg's square cross section
$x$ (centimeters)	$s (x)$ (centimeters)
0	6
10	5
20	4.5
30	4
40	5

Question 7 · from 8.7 · Calculate

No calculator. Use a right Riemann sum with the four subintervals of the table to approximate the volume of the leg, in cubic centimeters.

Your answer

Setup for question 8

An artist builds a clay sculpture on a flat base. The base is the region between the graph of a function $w$ and the $x$ -axis for $0 \leq x \leq 6$ , where $x$ and $w (x)$ are in centimeters, so the slice across the base at $x$ is $w (x)$ centimeters long. Cross sections of the sculpture perpendicular to the $x$ -axis are semicircles with diameters in the base. Some values of $w$ are in the table.

Length of the slice across the base
$x$ (centimeters)	$w (x)$ (centimeters)
0	2
2	4
4	8
6	6

Question 8 · from 8.8 · Calculate

No calculator. Use a right Riemann sum with the three subintervals the table gives to approximate the volume of the semicircle sculpture. The estimate is $k π$ cubic centimeters. What is $k$ ?

Your answer

Question 9 · from 8.9 · Fill the statement

No calculator. Tap a blank, then tap an answer.

To find the volume of a solid formed by revolving a region around the y-axis, cut the region into slices to the y-axis. Each disc's radius is then a width of the region, and the integral is taken with respect to .

Question 10 · from 8.10 · Sort it

No calculator. Sort each solid by the slices the disc method uses for it.

Discs of thickness $d x$

Discs of thickness $d y$

Question 11 · from 8.11 · Calculate

No calculator. The region bounded by $y = 5 - x$ , the line $y = 1$ and the $y$ -axis is revolved about the $x$ -axis. The cross section of the solid at $x = 1$ is a washer with area $k π$ square units. What is $k$ ?

Your answer

Question 12 · from 8.12 · Calculate

No calculator. The region in the first quadrant bounded by $y = x^{2}$ , the line $y = 4$ and the $y$ -axis is revolved about the line $x = 3$ . The volume of the solid is $k π$ . What is $k$ ?

Your answer

0 of 12 answered.