Calculus Homework Assignment

1. a. Use a Riemann sum with m = n = 2 to estimate the value of

\[ \iint_R xe^{-xy} dA \]

where R = [0, 2] × [0, 1]. Take the sample points to be upper right corners.

b. Use the Midpoint Rule to estimate the integral in part a.




2. Calculate the integral.

\[ {\bf a.}\; \int_0^2 \int_0^{\pi ∕ 2} x \sin y\, du\, dx.\]

\[ {\bf b.}\; \int_1^3 \int_1^2 \frac{1}{1 + x + y}\, dy\, dx. \]





3. Find the volume of the solid that lies under the hyperbolic paraboloid \( z = 3y^2 - x^2 + 2 \) and above the rectangle R = [–1, 1] × [1, 2].





4. Find the average value of

\[ f(x, y) = e^y\sqrt{x+ e^y} \]

over the rectangle R = [0, 4] × [0, 1].





5. Evaluate the double integral.

\[ \iint_D (x^2 + 2y) dA, \]

where D is bounded by y = x and \( y = x^3,\, x \geq 0. \)





6. Evaluate the integral by reversing the order of integration.

\[ \int_0^4 \int_{\sqrt{x}\,}^2 \frac{1}{y^3 + 1} dy dx \]




7. a. Use a double integral to find the area of the region, one loop of the
rose r = cos 3θ.

b. Use polar coordinates to find the volume of the solid above the cone

\[z = \sqrt{x^2 + y^2} \]

and below the sphere \( x^2 + y^2 + z^2 = 1. \)






8. Evaluate the iterated integral by converting to polar coordinates.

\[ \int_0^1 \int_y^{\sqrt{2 - y^2}} (x + y) dxdy \]