Posts tagged calculus 2
Surface area of revolution around the x-axis and y-axis

We can use integrals to find the surface area of the three-dimensional figure that’s created when we take a function and rotate it around an axis and over a certain interval. The formulas we use to find surface area of revolution are different depending on the form of the original function and the axis of rotation.

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Finding radius and interval of convergence of a Taylor series

Sometimes we’ll be asked for the radius and interval of convergence of a Taylor series. In order to find these things, we’ll first have to find a power series representation for the Taylor series.

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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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Using the comparison test to determine convergence or divergence

The comparison test for convergence lets us determine the convergence or divergence of the given series by comparing it to a similar, but simpler comparison series. We’re usually trying to find a comparison series that’s a geometric or p-series, since it’s very easy to determine the convergence of a geometric or p-series.

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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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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.

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The alternating series estimation theorem to estimate the value of the series and state the error

The alternating series estimation theorem gives us a way to approximate the sum of an alternating series with a remainder or error that we can calculate. To use the theorem, the alternating series must follow two rules.

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Probability density functions and probability of X in an interval

Probability density refers to the probability that a continuous random variable X will exist within a set of conditions. It follows that using the probability density equations will tell us the likelihood of an X existing in the interval [a,b].

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Geometric series test to figure out convergence

Before we can learn how to determine the convergence or divergence of a geometric series, we have to define a geometric series. Once you determine that you’re working with a geometric series, you can use the geometric series test to determine the convergence or divergence of the series.

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Finding the work required to empty a tank

To find the work required to empty a tank, first divide the tank into an infinite number of slices, then calculate the work required to remove a single slice of substance from the tank, then develop an equation to solve for the work needed to empty the entire tank, based on the work that was required to remove the single slice.

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Sum of the maclaurin series

One convenient way to find the sum of the Maclaurin series is to start with a well-known Maclaurin series and then manipulate it one step at a time until it matches the series you’ve been given. Because you’ll be manipulating the expression of the sum at the same time, once you get the series to match, you’ll automatically have the sum.

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Work done on elastic springs, and Hooke's law

To find the work required to stretch or compress an elastic spring, you’ll need to use Hooke’s Law. Every spring has its own spring constant k, and this spring constant is used in the Hooke’s Law formula.

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