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Calc 3 12.2

 

<== Calculus 3
Number Question Answer
See the work
1. Evaluate the iterated integral I = \int_{0}^{1}\!\!\int_{1-x}^{1+x} (18x^2 + 14y )\: dy dx

23 The work
2. Evaluate the iterated integral I = \int_{0}^{1}\!\!\int_{1-y}^{1+y} (3y^2 + 6x )\: dx dy

7.5 The work
3. Evaluate the double integral I = \iint_{\mathbf{D}} xy \, dA where D is the triangular region with vertices (0, 0), (5, 0), (0, 4).

50/3 The work
4. Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 2.

4/3 The work
5. Consider the integral \displaystyle \int_0^1 \int_{9 x}^{9}  f(x,y) dy dx. Sketch the region of integration and change the order of integration.

\displaystyle \int_a^b
\int_{g_1(y)}^{g_2(y)} f(x,y)  dx dy
a = b =
g_1(y) = g_2(y) =

a = 0
b = 9
0
y/9
The work
6. Consider the integral \displaystyle \int_0^1 \int_{9 x}^{9}  f(x,y) dy dx. Sketch the region of integration and change the order of integration.

\displaystyle \int_a^b
\int_{g_1(y)}^{g_2(y)} f(x,y)  dx dy
a = b =
g_1(y) = g_2(y) =

a = 0
b = 42
(y^2)/36
49
The work
7. Consider the integral \displaystyle \int_0^6 \int_0^{\sqrt{36-y}}  f(x,y) dx dy. If we change the order of integration we obtain the sum of two integrals:

\displaystyle \int_a^b
\int_{g_1(x)}^{g_2(x)} f(x,y)  dy dx + \displaystyle \int_c^d
\int_{g_3(x)}^{g_4(x)} f(x,y)  dy dx
a = b =
g_1(x) = g_2(x) =
c = d =
g_3(x) = g_4(x) =

a = 0
b = 5.477
0
6
c = 5.477
d = 6
0
36-x^2
The work
8. Consider the integral \displaystyle \int_0^1 \int_{9 x}^{9}  f(x,y) dy dx. Sketch the region of integration and change the order of integration.

\displaystyle \int_a^b
\int_{g_1(y)}^{g_2(y)} f(x,y)  dx dy
a = b =
g_1(y) = g_2(y) =

a = 0
b = 6.91

e^(y/3)
10
The work
9. In evaluating a double integral over a region, a sum of iterated integrals was obtained as follows:

\iint_D f(x,y)\, dA = \int_0^{4} \int_0^{y}  f(x,y)\, dx dy
+ \int_{4}^{8} \int_0^{8-y}  f(x,y)\, dx dy
\, .

Sketch the region D and express the double integral as an iterated integral with reversed order of integration.

\displaystyle \int_a^b
\int_{g_1(x)}^{g_2(x)} f(x,y)\,  dy dx
a = b =
g_1(x) = g_2(x) =
a = 0
b = 4

x
8-x
The work
10. Evaluate the integral by reversing the order of integration.
\int_{0}^{1}\!\!\int_{9y}^{9} e^{x^{2}} \, dx dy =
8.36*10^33 The work
11. Evaluate the integral by reversing the order of integration.
\int_{0}^{2}\!\!\int_{y^2}^{4} y \cos(x^2) \, dx dy =
-0.072 The work
12. Here is a more interesting problem to consider. We want to evaluate the improper integral

\int_0^\infty rac{	an^{-1}(7 x) - 	an^{-1}(5 x)}{x}\, dx.
Do it by rewriting the numerator of the integrand as \int_{f(x)}^{g(x)}
    h(y) dy for appropriate f, g, h and then reversing the order of integration in the resulting double integral.

The value of the improper intgeral is .

0.529 The work