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Calc 3 12.1

 

<== Calculus 3
Number Question Answer
See the work
1. Consider the solid that lies above the square (in the xy-plane) R = [0, 2] 	imes [0, 2],
and below the elliptic paraboloid z = 25 - x^{2} - 3y^2.

(A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners.

(B) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners..

(C) What is the average of the two answers from (A) and (B)?

A) 92
B) 60
C) 76
The work
2. Consider the solid that lies above the square (in the xy-plane) R = [0, 1] 	imes [0, 1],
and below the elliptic paraboloid z = 81 - x^{2} +5 xy - 4y^2.

Estimate the volume by dividing R into 9 equal squares and choosing the sample points to lie in the midpoints of each square.

80.63 The work
3. Using geometry, calculate the volume of the solid under z = \sqrt{ 9 - x^{2} - y^{2} } and over the circular disk x^{2} + y^{2} \leq 9

56.549 The work
4. Evaluate the iterated integral \int_{0}^{1} \int_{0}^{2} 3x^2y^3 \, dx dy

2 The work
5. valuate the iterated integral \int_{1}^{2} \int_{3}^{4} (3x + y)^{-2} \: dy dx

0.01626 The work
6. Evaluate the integral \int_0^{\pi/4} \int_{5}^{7}( y\cos x  - 2) \,dy dx.
5.344 The work
7. Find \int_{0}^{5} \int_{0}^{3} xy e^{x+y} \,dydx
24482.49 The work
8. Find \int_{3}^{5} \int_{7}^{9}( x + \ln y) \,dydx
24.307 The work
9. Find \iint_R f(x,y)\, dA where f(x,y) = x and R = [3, 9] 	imes [5, 10].
\iint_R f(x,y)\, dA = =
180 The work
10. Find \iint_R f(x,y)\, dA where f(x,y) = 2 x + 5 and R = [1, 4] 	imes [-4, -2].
\iint_R f(x,y)\, dA = =
60 The work
11. Calculate the double integral \int \int_{\mathbf{R}} (8x + 4y + 32 )\: dA where R is the region: 0 \leq x \leq 2, 0 \leq y \leq 4.

384 The work
12. Calculate the double integral \int \int_{\mathbf{R}} x \cos(2x + y) \: dA where R is the region: 0 \leq x \leq rac{\pi}{6}, 0 \leq y \leq rac{\pi}{4}

0.0469 The work
13. Consider the solid that lies above the square (in the xy-plane) R = [0, 2] 	imes [0, 2],
and below the elliptic paraboloid z = 25 - x^{2} - 4y^2.

Using iterated integrals, compute the exact value of the volume.

220/3 The work
14. Calculate the volume under the elliptic paraboloid z = 4x^2 + 5y^2 and over the rectangle R = [-4, 4] 	imes [-2, 2].

896 The work
15. Find the average value of f(x,y) = 4 e^{ y} \sqrt{x+e^{ y}} over the rectangle R = [0,1] 	imes [0,4].
Average value =
271.42 The work
16. Find the average value of f(x,y) = 4 x^4 y^4 over the rectangle R with vertices (-3,0), (-3,1), (3,0), (3,1).
Average value =
12.96 The work
17. If \displaystyle \int_{5}^{7} f(x) \: dx = -1 and \displaystyle \int_{2}^{5} g(x) \: dx = -3, what is the value of \displaystyle \int\!\!\int_{D} f(x)\!g(y) \: dA where D is the square: 5 \leq x \leq 7, \ \ 2 \leq y \leq 5?

3 The work