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Math 21D Practice Midterm
创建日期：2024-09-05 15:38:38
所属分类：数学mathematics
标签： Math 21D
Practice Midterm

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Math 21D

Practice Midterm #1

1. Integrate the function f(x, y, z) = 3x 2yz over the region described by the inequalities −1 ≤ x ≤ 2, 0 ≤ z ≤ 1, 0 ≤ y ≤ 1.

2. Consider a triangular plate with vertices at (0, 0), (3, 0), and (3, 6), whose density at (x, y) is given by δ(x, y) = 3x + 2y.

a. Set up (but do not compute) an iterated integral giving the plate’s area.

b. Find the plate’s mass.

c. Find the coordinates of the plate’s center of mass.

3. Consider the region R in three-dimensional space bounded below by the xy-plane, on the side by the surface y = 1 − x 2 , and on top by the plane z = 2y. A picture is provided below:

a. Set up a triple integral to compute the volume of R using the integration order dz dy dx.

b. Set up a triple integral to compute the volume of R using the integration order dy dz dx.

c. Compute the volume of R using any integration order you like.

4. For each part, set up an integral to compute the area or volume of the region. You may use any coordinate system you like, in any integration order you like.

a. The region in the first quadrant between the circles x2 + y2 = 4 and x2 + y2 = 9.

b. The solid bounded below by the xy-plane, on the sides by the sphere x2 + y2 + z2 = 2, and above by the cone

c. The solid bounded by the xy-plane, the surface z = 1 − y2 , and the planes x = 0, x = 1.

5. Convert the following integral to cylindrical coordinates, and evaluate.

6. Let R be the region in two-dimensional space bounded by x + y = 3, x − y = −2, x + y = 6, x − y = 1.

Evaluate the following integral by choosing an appropriate substitution.

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