Problem 8
Evaluate the integral and subsequent iterated integral. (a) \(\int_{y}^{y^{2}}(x-y) d x\) (b) \(\int_{-1}^{1} \int_{y}^{y^{2}}(x-y) d x d y\)
Problem 8
Set up the iterated integral that computes the surface area of the given surface over the region \(R .\) \(f(x, y)=\frac{1}{x^{2}+y^{2}+1} ; \quad R\) is bounded by the circle \(x^{2}+\) \(y^{2}=9\).
Problem 8
(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration. $$ \int_{1}^{3} \int_{y}^{3}\left(x^{2} y-x y^{2}\right) d x d y $$
Problem 9
Point masses are given along a line or in the plane. Find the center of \(\operatorname{mass} \bar{x}\) or \((\bar{x}, \bar{y}),\) as appropriate. (All masses are in grams and distances are in cm.) $$ \begin{array}{l} m_{1}=2 \text { at }(-2,-2) ; \quad m_{2}=2 \text { at }(2,-2) ; \\ m_{3}=20 \text { at }(0,4) \end{array} $$
Problem 9
Evaluate the integral and subsequent iterated integral. (a) \(\int_{0}^{y}(\cos x \sin y) d x\) (b) \(\int_{0}^{\pi} \int_{0}^{y}(\cos x \sin y) d x d y\)
Problem 9
Set up the triple integrals that give the volume of \(D\) in all 6 orders of integration, and find the volume of \(D\) by evaluating the indicated triple integral. \(D\) is bounded by the coordinate planes and \(z=2-2 x / 3-2 y\).
Problem 9
In Exercises \(3-10\), a function \(f(x, y)\) is given and a region \(R\) of the \(x-y\) plane is described. Set up and evaluate \(\iint_{R} f(x, y) d A\) using polar coordinates. \(f(x, y)=x^{2}-y^{2} ; R\) is the region enclosed by the circle \(x^{2}+y^{2}=36\) in the first and fourth quadrants.
Problem 9
(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration. $$ \int_{0}^{1} \int_{-\sqrt{1-y}}^{\sqrt{1-y}}(x+y+2) d x d y $$
Problem 9
Set up the iterated integral that computes the surface area of the given surface over the region \(R .\) \(f(x, y)=x^{2}-y^{2} ; \quad R\) is the rectangle with opposite corners (-1,-1) and (1,1).
Problem 10
Set up the triple integrals that give the volume of \(D\) in all 6 orders of integration, and find the volume of \(D\) by evaluating the indicated triple integral. \(D\) is bounded by the planes \(y=0, y=2, x=1, z=0\) and \(z=(3-x) / 2\) Evaluate the triple integral with order \(d x d y d z\).
