Problem 4
In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{\cos y} y d x $$
Problem 4
Evaluate the iterated integral. $$ \int_{1}^{4} \int_{1}^{e^{2}} \int_{0}^{1 / x z} \ln z d y d z d x $$
Problem 4
In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u v-2 u, y=u v\)
Problem 4
Find the area of the surface given by \(z=f(x, y)\) over the region \(R .\) $$ \begin{array}{l} f(x, y)=2+\frac{2}{3} y^{3 / 2} \\ R=\\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 2-x\\} \end{array} $$
Problem 4
Approximate the integral \(\int_{R} \int f(x, y) d A\) by dividing the rectangle \(R\) with vertices \((0,0),\) \((4,0),(4,2),\) and (0,2) into eight equal squares and finding the sum \(\sum_{i=1}^{8} f\left(x_{i}, y_{i}\right) \Delta A_{i}\) where \(\left(x_{i}, y_{i}\right)\) is the center of the \(i\) th square. Evaluate the iterated integral and compare it with the approximation. $$ \int_{0}^{4} \int_{0}^{2} \frac{1}{(x+1)(y+1)} d y d x $$
Problem 4
Evaluate the iterated integral. $$ \int_{0}^{\pi / 4} \int_{0}^{\pi / 4} \int_{0}^{\cos \theta} \rho^{2} \sin \phi \cos \phi d \rho d \theta d \phi $$
Problem 5
Sketch the region \(R\) and evaluate the iterated integral \(\int_{R} \int f(x, y) d A .\) $$ \int_{0}^{2} \int_{0}^{1}(1+2 x+2 y) d y d x $$
Problem 5
In Exercises 5 and 6 , sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. $$ \int_{0}^{2 \pi} \int_{0}^{\sqrt{3}} \int_{0}^{3-r^{2}} r d z d r d \theta $$
Problem 5
In Exercises \(5-10,\) evaluate the double integral \(\int_{R} \int f(r, \theta) d A,\) and sketch the region \(R\). $$ \int_{0}^{2 \pi} \int_{0}^{6} 3 r^{2} \sin \theta d r d \theta $$
Problem 5
Evaluate the iterated integral. $$ \int_{0}^{4} \int_{0}^{\pi / 2} \int_{0}^{1-x} x \cos y d z d y d x $$
