91Ó°ÊÓ

Convert the integrals to polar coordinates and evaluate. $$\int_{0}^{\sqrt{6}} \int_{-x}^{x} d y d x$$

Let \(p\) be the joint density function such that \(p(x, y)=x y\) in \(R,\) the rectangle \(0 \leq x \leq 2,0 \leq y \leq 1\) and \(p(x, y)=0\) outside \(R .\) Find the fraction of the population satisfying the given constraints. $$x+y \leq 1$$

Let \(W\) be the solid cone bounded by \(z=\sqrt{x^{2}+y^{2}}\) and \(z=2.\) Decide (without calculating its value) whether the integral is positive, negative, or zero. $$\int_{W} y d V$$

Convert the integrals to polar coordinates and evaluate. \(\int_{0}^{\sqrt{2}} \int_{y}^{\sqrt{4-y^{2}}} x y d x d y\)

Explain what is wrong with the statement. For all \(f,\) the integral \(\int_{R} f(x, y) d A\) gives the volume of the solid under the graph of \(f\) over the region \(R\)

Let \(p\) be the joint density function such that \(p(x, y)=x y\) in \(R,\) the rectangle \(0 \leq x \leq 2,0 \leq y \leq 1\) and \(p(x, y)=0\) outside \(R .\) Find the fraction of the population satisfying the given constraints. $$0 \leq x \leq 1,0 \leq y \leq 1 / 2$$

Write a triple integral in cylindrical coordinates giving the volume of a sphere of radius \(K\) centered at the origin. Use the order \(d z d r d \theta\)

Let \(W\) be the solid cone bounded by \(z=\sqrt{x^{2}+y^{2}}\) and \(z=2.\) Decide (without calculating its value) whether the integral is positive, negative, or zero. $$\int_{W} x d V$$

Write a triple integral in spherical coordinates giving the volume of a sphere of radius \(K\) centered at the origin. Use the order \(d \theta\) d \(\rho d \phi\)

Give an example of: A function \(f(x, y)\) and rectangle \(R\) such that the Riemann sums obtained using the lower left-hand corner of each subrectangle are an overestimate.