Problem 19
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{2} \int_{0}^{\sqrt{4-y^{2}}} \frac{2}{\sqrt{4-y^{2}}} d x d y $$
Problem 19
Problem 19
In Exercises \(19-22,\) use polar coordinates to set up and evaluate the double integral \(\int_{R} \int f(x, y) d A\). $$ f(x, y)=x+y, R: x^{2}+y^{2} \leq 4, x \geq 0, y \geq 0 $$
Problem 20
Use polar coordinates to set up and evaluate the double integral \(\int_{R} \int f(x, y) d A\). $$ f(x, y)=e^{-\left(x^{2}+y^{2}\right) / 2}, R: x^{2}+y^{2} \leq 25, x \geq 0 $$
Problem 20
Use cylindrical coordinates to verify the given formula for the moment of inertia of the solid of uniform density. Right circular cylinder: \(I_{z}=\frac{3}{2} m a^{2}\) \(r=2 a \sin \theta, \quad 0 \leq z \leq h\) Use a computer algebra system to evaluate the triple integral.
Problem 20
List the six possible orders of integration for the triple integral over the solid region \(Q\) \(\iint_{Q} \int x y z d V\). $$ Q=\left\\{(x, y, z): 0 \leq x \leq 1, y \leq 1-x^{2}, 0 \leq z \leq 6\right\\} $$
Problem 20
Use a computer algebra system to approximate the double integral that gives the surface area of the graph of \(f\) over the region \(R=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\\}\) $$ f(x, y)=\frac{2}{5} y^{5 / 2} $$
Problem 20
In Exercises 17-22, use a change of variables to find the volume of the solid region lying below the surface \(z=f(x, y)\) and above the plane region \(R\). \(f(x, y)=(3 x+2 y)(2 y-x)^{3 / 2}\) \(R:\) region bounded by the parallelogram with vertices \((0,0),\) (-2,3),(2,5),(4,2)
Problem 20
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{\pi / 2} \int_{0}^{2 \cos \theta} r d r d \theta $$
Problem 21
In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{\pi / 2} \int_{0}^{\sin \theta} \theta r d r d \theta $$
