Finding the Volume Introduction,Examples with Step-by-Step Explanations

The problems about Finding the Volume

Topic	Problem	Solution
None	Let $R$ be the region bounded by the fol…	Given the curve $y = \cos 2 x$ , the x-axis, and the y-axis, we need to find the volume of the soli…
None	1 Let $R$ be the region bounded by the f…	Let $R$ be the region bounded by the following curves. Find the volume of the solid gene…
None	Question Let $f (x) = - \frac{x^{3}}{2}$ over the int…	We are given the function $f (x) = - \frac{x^{3}}{2}$ and we are asked to find the surface area of …
None	Question Set up an integral that represents the s…	Given the function $f (x) = \sqrt{5 x - 8}$ , we are asked to find the surface area of the surface g…
None	Suppose that $f$ is a continuous function such th…	Let's use the substitution method to solve the integral. We can let $u = \sqrt{x}$ , then \(du = \…
None	Find the area of the surface obtained by rotating…	The formula for the surface area of a curve $y = f (x)$ , $a \leq x \leq b$ , rotated about the y-axis i…
None	Find $\iint_{D} (2 x + y) d A$ where $D=\left\{(x, y…	The given region D is a semi-circle with radius 4 in the first quadrant. To solve this double integ…
None	Use the shell method to find the volume generated…	The shaded region can be divided into two parts: a rectangle with a height of 5 units and a width o…
None	Evaluate $\iint y e^{x y} d A$ for the region $R=…	We are given the double integral $\iint y e^{x y} d A$ over the region \(R=\{(x, y) \mid 0 \leq x…
None	a. Write the integral that gives the area of the …	First, we need to understand the problem. We are asked to find the surface area generated when the …
None	Determine the volume of the solid in the 1 st oct…	The given equation is of a plane in 3D space. The plane intersects the x, y, and z axes at \((1,0,0…
None	Find $∭_{E} y d V$ , where $E$ is the solid b…	First, we need to find the limits of integration. The solid E is bounded by the parabolic cylinder …
None	1. Find the volume of the solid that lies under t…	The volume of the solid under the surface $z = f (x, y)$ and above the region $D$ in the $x y$ -pl…
None	Find the volume when the region bounded by the $y…	The volume of a solid of revolution can be found using the formula for the volume of a disk or wash…
None	A trough is 2 feet long and 1 foot high. The vert…	The work required to pump the water out of the trough is equal to the weight of the water times the…
None	The volume of the solid obtained by rotating the …	The volume of a solid obtained by rotating a region about a line can be computed using the method o…
None	Find the volume of the solid formed by rotating t…	First, we need to understand that the solid is formed by rotating the region enclosed by the given …
None	A clothing company determines that its marginal c…	The marginal cost function, $C^{'} (x)$ , gives the cost to produce one additional dress. To find the …
None	Considere o campo vetorial \[ \mathbf{F}(x, y, z)…	Consider the vector field \(\mathbf{F}(x, y, z)=\left(\frac{x^{3}}{3}+x y^{2}\right) \mathbf{i}+\le…
None	Let the region $R$ be the area enclosed …	Find the intersection points of $f (x) = 2 \sqrt{x}$ and $g (x) = 2 x$ to determine the limits of i…
None	Let the region $R$ be the area enclosed …	Find the points of intersection between $f (x) = 3 x^{2}$ and $g (x) = 4 x$ by setting $f (x) = g (x)$
None	Let the region $R$ be the area enclosed …	Find the intersection points of $f (x) = e^{x}$ and $y = 3$ to determine the limits of integration…
None	Let the region $R$ be the area enclosed …	Find the intersection points of the function $f (x) = x^{2} - 2$ and the horizontal line $y = 6$ t…
None	Find the surface area of the surface generated by…	\int_{0}^{1} 2\pi y \sqrt{1+{\left( \frac{dy}{dx} \right)}^2} dx
None	$D$ is the solid inside the cylinder \( x^{2}…	Define cylindrical coordinates transformation: $(x, y, z) = (r \cos θ, r \sin θ, z)$ wi…
None	Question 1 (7 points) Find the volume of the soli…	$V = π \int_{0}^{2} (5 - x^{2})^{2} - (5)^{2} d x$
None	Question 21 (5.2 points) Let $R$ be the regio…	$V = π \int_{0}^{4} (\sqrt{x})^{2} d x$