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Area And Volume Revisited | Calculus - Mathematics PDF Download
This section is here only so we can summarize the geometric interpretations of the double and triple integrals that we saw in this chapter. Since the purpose of this section is to summarize these formulas we aren’t going to be doing any examples in this section.
We’ll first look at the area of a region. The area of the region D is given by
Now let’s give the two volume formulas. First the volume of the region E is given by
Finally, if the region E can be defined as the region under the function z = f ( x , y ) and above the region D in x y -plane then,
Note as well that there are similar formulas for the other planes. For instance, the volume of the region behind the function y = f ( x , z ) and in front of the region D in the x z -plane is given by,
Likewise, the the volume of the region behind the function x = f ( y , z ) and in front of the region D in the y z -plane is given by,
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FAQs on Area And Volume Revisited - Calculus - Mathematics
| 1. What is the formula for finding the area of a rectangle? |  |
Ans. The formula for finding the area of a rectangle is length multiplied by width. In mathematical terms, it can be expressed as A = l * w, where A represents the area, l represents the length, and w represents the width.
| 2. How do you calculate the volume of a cylinder? | |
Ans. The volume of a cylinder can be calculated using the formula V = π * r^2 * h, where V represents the volume, π is a mathematical constant (approximately equal to 3.14159), r is the radius of the circular base, and h is the height of the cylinder.
| 3. Can you provide an example of finding the area of a triangle? | |
Ans. Certainly! Let's say we have a triangle with a base of 6 units and a height of 4 units. To find the area of this triangle, we can use the formula A = 0.5 * b * h, where A represents the area, b is the base, and h is the height. Plugging in the values, we get A = 0.5 * 6 * 4 = 12 square units.
| 4. How can I calculate the volume of a sphere? | |
Ans. To calculate the volume of a sphere, you can use the formula V = (4/3) * π * r^3, where V represents the volume, π is a mathematical constant (approximately equal to 3.14159), and r is the radius of the sphere.
| 5. What is the formula for finding the area of a circle? | |
Ans. The formula for finding the area of a circle is A = π * r^2, where A represents the area and r is the radius of the circle.
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