Applications of Integration unit test
A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.
When the amount is greatest
Cross-sectional side from the base
In minus out
Perpendicular to which axis
Horizontal slices mean right minus left
Rectangle cross section with a stated ratio
Area between two curves
Volume by disks
More than two intersections
Mean Value Theorem for integrals
Volume by washers
Area is never negative
Short answer 1. Define or explain: Integrating with respect to y
3 ptsShort answer 2. Define or explain: Cross-section area formulas
3 ptsShort answer 3. Define or explain: Volume of revolution about a horizontal line
3 ptsShort answer 4. Define or explain: Washer inner and outer radii
3 ptsFree response
9 ptsCALCULATOR PERMITTED. The shaded region R is bounded by the graphs of the functions f and g, where f(x) = x² − 2x and g(x) = x + sin(πx). The graphs intersect at x = 0 and x = 3, with g(x) ≥ f(x) on [0, 3], and R is the region between them. (Your calculator should be in radian mode.)
A. Find the area of R. Show the setup for your calculations.
B. Region R is the base of a solid. For this solid, at each x the cross section perpendicular to the x-axis is a rectangle with height x and base in region R. Find the volume of the solid. Show the setup for your calculations.
C. Write, but do not evaluate, an integral expression for the volume of the solid generated when the region R is rotated about the horizontal line y = −2.
D. It can be shown that g′(x) = 1 + π·cos(πx). Find the value of x, for 0 < x < 1, at which the line tangent to the graph of f is parallel to the line tangent to the graph of g.