ConcepTest Section 16.3 Question 1 What does the integral represent? (a) The volume of a cube of side 1. (b) The volume of a sphere of radius 1. (c) The.

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Presentation transcript:

ConcepTest Section 16.3 Question 1 What does the integral represent? (a) The volume of a cube of side 1. (b) The volume of a sphere of radius 1. (c) The area of a square of side 1. (d) None of the above.

ConcepTest Section 16.3 Answer 1 (d) None of the above. ANSWER COMMENT: Follow-up Question. What could it represent?

ConcepTest Section 16.3 Question 2 Set up a single, double, and triple integral, each of which gives the volume of the region under the plane z = x and above the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2. Explain what each integrand represents.

ConcepTest Section 16.3 Answer 2 here, 2x is the area of a rectangular cross section of the three dimensional region parallel to the yz-plane. Then 2xdx is the volume of a slab parallel to the yz-plane, but only infinitesimally thick in the x- direction. here x is the length of a vertical line segment from the bottom to the top of the region. Then, xdydx is the volume of a vertical column that is infinitesimally wide in the x- and y-directions. here the entire region is described by the limits of integration. Then, dzdydx represents the volume of an infinitesimal box anywhere in the region. ANSWER COMMENT: If you start with the triple integral, the other integrals are stages of the iterative solution.

ConcepTest Section 16.3 Question 3 For each integral (a) – (d) that makes sense, match it with its region of integration, I or II. I The region below the plane z = y above the triangle with vertices (0, 1), (2, 1), (2, 3) in the xy-plane. II The region between the upper hemisphere of x 2 + y 2 + z 2 = 1 and the xy-plane.

ConcepTest Section 16.3 Answer 3 I is the region of integration of the first integral, (a). II is the region of integration of the last two integrals, (c) and (d). ANSWER COMMENT: Follow-up Question. What is wrong with the second integral?

ConcepTest Section 16.3 Question 4 Which of the following integrals is (are) equal to

ConcepTest Section 16.3 Answer 4 ANSWER COMMENT: Follow-up Question. What is the region of integration of the other integrals?

ConcepTest Section 16.3 Question 5 describes the mass of (a) A cone that gets heavier toward the outside. (b) A cone that gets lighter toward the outside. (c) A ball that gets heavier toward the outside. (d) A ball that gets lighter toward the outside.

ConcepTest Section 16.3 Answer 5 (c) A ball that gets heavier toward the outside. ANSWER COMMENT: Follow-up Question. How would the integral have to change to represent each of the other choices?