The concept of Volume refers to the quantity of space that a three-dimensional object takes up, considering its length, breadth, and height. This is usually quantified in cubic units. The method to calculate volume can vary based on the object's shape. For instance, to ascertain the volume of a cube, one would multiply its length, breadth, and height together.
| Topic | Problem | Solution |
|---|---|---|
| None | Let $\mathrm{R}$ be the region bounded by the fol… | Given the curve \(y = \cos 2x\), the x-axis, and the y-axis, we need to find the volume of the soli… |
| None | 1 Let $\mathrm{R}$ be the region bounded by the f… | Let \(\mathrm{R}\) 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 \(f(x) = \sqrt{5x - 8}\), 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 $\iiint_{E} y d V$, 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 $\mathrm{R}$ be the area enclosed … | Find the intersection points of \(f(x) = 2\sqrt{x}\) and \(g(x) = 2x\) to determine the limits of i… |
| None | Let the region $\mathrm{R}$ be the area enclosed … | Find the points of intersection between \(f(x) = 3x^2\) and \(g(x) = 4x\) by setting \(f(x) = g(x)\) |
| None | Let the region $\mathrm{R}$ 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 $\mathrm{R}$ 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: \((x, y, z) = (r \cos \theta, r \sin \theta, z)\) wi… |
| None | Question 1 (7 points) Find the volume of the soli… | \(V = \pi \int_0^2 (5-x^2)^2 - (5)^2 dx\) |
| None | Question 21 (5.2 points) Let \( R \) be the regio… | \( V = \pi \int_{0}^{4} (\sqrt{x})^2 dx \) |