91Ó°ÊÓ

Sketch the region of integration. $$\int_{0}^{\pi / 4} \int_{0}^{1 / \cos \theta} f(r, \theta) r d r d \theta$$

A joint probability density function is given by \(p(x, y)=0.005 x+0.025 y\) in \(R,\) the rectangle \(0 \leq x \leq 10,0 \leq y \leq 2,\) and \(p(x, y)=0\) else. Find the probability that a point \((x, y)\) satisfies the given conditions. $$x \leq 4 \text { and } y \geq 1$$

Sketch the region of integration. $$\int_{3}^{4} \int_{3 \pi / 4}^{3 x / 2} f(r, \theta) r d \theta d r$$

A joint probability density function is given by \(p(x, y)=0.005 x+0.025 y\) in \(R,\) the rectangle \(0 \leq x \leq 10,0 \leq y \leq 2,\) and \(p(x, y)=0\) else. Find the probability that a point \((x, y)\) satisfies the given conditions. $$x \geq 5 \text { and } y \geq 1$$

For \(x, y\) and \(z\) in meters, what does the integral over the solid region \(E\) represent? Give units. $$\int_{E} 1 d V$$

For Exercises \(13-20,\) sketch the region of integration and evaluate the integral. $$\int_{0}^{2} \int_{0}^{x} e^{x^{2}} d y d x$$

Decide (without calculation) whether the integrals are positive, negative, or zero. Let \(D\) be the region inside the unit circle centered at the origin, let \(R\) be the right half of \(D\), and let \(B\) be the bottom half of \(D\) $$\int_{B}\left(y-y^{3}\right) d A$$

For \(x, y\) and \(z\) in meters, what does the integral over the solid region \(E\) represent? Give units. \(\int_{E} \delta(x, y, z) d V,\) where \(\delta(x, y, z)\) is density, in \(\mathrm{kg} / \mathrm{m}^{3}.\)

For Exercises \(13-20,\) sketch the region of integration and evaluate the integral. $$\int_{1}^{5} \int_{x}^{2 x} \sin x d y d x$$

Sketch the region of integration. $$\int_{x / 4}^{\pi / 2} \int_{0}^{2 / \sin \theta} f(r, \theta) r d r d \theta$$