Posts tagged calculus iii
Finding volume for triple integrals using spherical coordinates

We can use triple integrals and spherical coordinates to solve for the volume of a solid sphere. To convert from rectangular coordinates to spherical coordinates, we use a set of spherical conversion formulas.

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Finding gradient vectors for multivariable functions

The gradient vector formula gives a vector-valued function that describes the function’s gradient everywhere. If we want to find the gradient at a particular point, we just evaluate the gradient function at that point.

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Finding surface area of one function, bounded by another function, using a double integrals

You can use a double integral to find the area of a surface, bounded by another surface. The most difficult part of this will be finding the bounds of each of the integrals in the double integral.

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Solving triple integrals using the midpoint rule

Similarly to the way we used midpoints to approximate single integrals by taking the midpoint at the top of each approximating rectangle, and to the way we used midpoints to approximate double integrals by taking the midpoint at the top of each approximating prism, we can use midpoints to approximate a triple integral by taking the midpoint of each sub-cube.

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How to calculate the differential of any multivariable function

Before we can use the formula for the differential, we need to find the partial derivatives of the function with respect to each variable. Then the differential for a multivariable function is given by three separate formulas.

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Cylindrical coordinates and triple integrals

Like cartesian (or rectangular) coordinates and polar coordinates, cylindrical coordinates are just another way to describe points in three-dimensional space. Cylindrical coordinates are exactly the same as polar coordinates, just in three-dimensional space instead of two-dimensional space.

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Six ways to write the same iterated triple integral

There are six ways to express an iterated triple integral. While the function inside the integral always stays the same, the order of integration will change, and the limits of integration will change to match the order.

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Evaluating double integrals as iterated integrals

Whenever we’re given a double integral, we want to turn it into an iterated integral, because with iterated integrals, we can easily evaluate one integral at a time, like we would in single variable calculus. When we evaluate iterated integrals, we always work from the inside out.

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