R= [0,6] x [0, 2]. Take sample points to be the lov 2. Compute /. Yev? 1+ x² dx dy. 3. Compute 3ry? dy dx.
R= [0,6] x [0, 2]. Take sample points to be the lov 2. Compute /. Yev? 1+ x² dx dy. 3. Compute 3ry? dy dx.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
please do question 2
![1. Use a Riemann sum with \( m = 3 \) and \( n = 2 \) to estimate the value of
\[
\iint_R (x + 2y) \, dA
\]
where \( R = [0, 6] \times [0, 2] \). Take sample points to be the lower right corners.
2. Compute
\[
\int_{-1}^2 \int_0^1 \frac{ye^{y^2}}{1 + x^2} \, dx \, dy.
\]
3. Compute
\[
\int_0^1 \int_x^{e^x} 3xy^2 \, dy \, dx.
\]
4. Compute
\[
\int_0^1 \int_0^y \int_0^x 6xyz \, dz \, dx \, dy.
\]
5. Compute
\[
\int_0^1 \int_x^1 \cos(y^2) \, dy \, dx
\]
by reversing the order of integration.
6. Find the volume of the solid bounded by the paraboloids \( z = x^2 + y^2 \) and \( z = 2 - x^2 - y^2 \).
7. Compute
\[
\int_0^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \frac{xy}{x^2 + y^2} \, dy \, dx
\]
by converting to polar coordinates.
8. Find the \( x \)-coordinate of the center of mass of the lamina that occupies the region
\[
D = \{(x, y) \, | \, 0 \leq x \leq 1, \, x^2 \leq y \leq 1\}
\]
and has density function \( \rho(x, y) = x + y \).
9. Find the surface area of the part of the cylinder \( y^2 + z^2 = 9 \) that is above the rectangle
\[
R = [0, 2] \times [-3,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe51bf8eb-f11a-41aa-abbd-22dd34414af1%2Fedc5a1c3-6eff-48de-b023-f8bbe6a0d875%2F978izta_processed.png&w=3840&q=75)
Transcribed Image Text:1. Use a Riemann sum with \( m = 3 \) and \( n = 2 \) to estimate the value of
\[
\iint_R (x + 2y) \, dA
\]
where \( R = [0, 6] \times [0, 2] \). Take sample points to be the lower right corners.
2. Compute
\[
\int_{-1}^2 \int_0^1 \frac{ye^{y^2}}{1 + x^2} \, dx \, dy.
\]
3. Compute
\[
\int_0^1 \int_x^{e^x} 3xy^2 \, dy \, dx.
\]
4. Compute
\[
\int_0^1 \int_0^y \int_0^x 6xyz \, dz \, dx \, dy.
\]
5. Compute
\[
\int_0^1 \int_x^1 \cos(y^2) \, dy \, dx
\]
by reversing the order of integration.
6. Find the volume of the solid bounded by the paraboloids \( z = x^2 + y^2 \) and \( z = 2 - x^2 - y^2 \).
7. Compute
\[
\int_0^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \frac{xy}{x^2 + y^2} \, dy \, dx
\]
by converting to polar coordinates.
8. Find the \( x \)-coordinate of the center of mass of the lamina that occupies the region
\[
D = \{(x, y) \, | \, 0 \leq x \leq 1, \, x^2 \leq y \leq 1\}
\]
and has density function \( \rho(x, y) = x + y \).
9. Find the surface area of the part of the cylinder \( y^2 + z^2 = 9 \) that is above the rectangle
\[
R = [0, 2] \times [-3,
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