U-substitution to solve definite integrals

U-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. If you don’t change the limits of integration, then you’ll need to back-substitute for the original variable at the end.

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Polar coordinates vs. rectangular coordinates

Any point in the coordinate plane can be expressed in both rectangular coordinates and polar coordinates. Instead of moving out from the origin using horizontal and vertical lines, like we would with rectangular coordinates, in polar coordinates we instead pick the angle, which is the direction, and then move out from the origin a certain distance.

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Vertical angles as congruent angles

Vertical angles are angles in opposite corners of intersecting lines. So vertical angles always share the same vertex, or corner point of the angle. They’re a special angle pair because their measures are always equal to one another, which means that vertical angles are congruent angles.

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Theorem of Pappus to find volume using the centroid

The Theorem of Pappus tells us that the volume of a three-dimensional solid object that’s created by rotating a two-dimensional shape around an axis is given by V=Ad. V is the volume of the three-dimensional object, A is the area of the two-dimensional figure being revolved, and d is the distance traveled by the centroid of the two-dimensional figure.