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### How to Approximate Area with Midpoint Rectangles

A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangle's top side*.* A midpoint sum is a much better estimate of area than either a left-rectangle [more…]

### How to Approximate Area with the Trapezoid Rule

With the trapezoid rule, instead of approximating area by using rectangles (as you do with the left, right, and midpoint rectangle methods), you approximate area with — can you guess? — trapezoids. [more…]

### How to Approximate Area with Simpson's Rule

With Simpson’s rule, you approximate the area under a curve with curvy-topped “trapezoids.” The tops of these shapes are sections of parabolas. You can call them “trapezoids” because they play the same [more…]

### How to Do Integration by Parts

Integrating by parts is the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you [more…]

### How to Solve a Quadratic Equation by Completing the Square

You can solve quadratic equations by *completing the square*. Completing the square involves creating a perfect square trinomial from the quadratic equation, and then solving that trinomial by taking its [more…]

### How to Use Sigma Notation

For adding up long series of numbers like the rectangle areas in a left, right, or midpoint sum, sigma notation comes in handy. Here’s how it works. Say you wanted to add up the first 100 multiples of [more…]

### How to Do Integration by Parts More than Once

Sometimes you have to use the integration-by-parts method more than once because the first run through the method takes you only part way to the answer. [more…]

### How to Calculate Instantaneous Speed with Limits

You can calculate the instantaneous speed of an object using limits. Say that you and your calculus-loving cat are hanging out one day, and you decide to drop a ball out of your second-story window. Here’s [more…]

### How to Know When a Derivative Doesn't Exist

There are three situations where a derivative fails to exist. The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s [more…]

### How to Use the Chain Rule to Find the Derivative of Nested Functions

Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another [more…]

### How to Find Local Extrema with the First Derivative Test

All local maximums and minimums on a function's graph — called local extrema — occur at critical points of the function (where the derivative is zero or undefined). [more…]

### How to Find Absolute Extrema on a Closed Interval

Every function that’s continuous on a closed interval has an absolute maximum value and an absolute minimum value (the absolute extrema) in that interval — in other words, a highest and lowest point — [more…]

### How to Locate Intervals of Concavity and Inflection Points

You can locate a function's concavity (where a function is concave up or down) and inflection points (where the concavity switches from positive to negative or vice versa) in a few simple steps. The following [more…]

### How to Make Linear Approximations

Because ordinary functions are locally *linear* (that means straight) — and the further you zoom in on them, the straighter they look—a line tangent to a function is a good approximation of the function [more…]

### How Integration Works: It’s Just Fancy Addition

The most fundamental meaning of integration is to add up. And when you depict integration on a graph, you can see the adding up process as a summing up of thin rectangular strips of area to arrive at the [more…]

### How to Approximate Area with Right Rectangles

You can approximate the area under a curve by adding up “right” rectangles. This method works just like the left sum method except that each rectangle is drawn so that its right upper corner touches the [more…]

### How to Find Antiderivatives with Reverse Rules

The easiest antiderivative rules are the ones that are simply the reverse of derivative rules that you probably already know. These rules are automatic, one-step antiderivatives, with the exception of [more…]

### How to Find Antiderivatives by Guessing and Checking

The guess-and-check method works when the *integrand*— that’s the thing you want to antidifferentiate (the expression after the integral symbol, not counting the [more…]

### How to Find Antiderivatives with the Substitution Method

When a function’s argument (that’s the function’s input) is more complicated than something like 3*x* + 2 (a *linear* function of *x* — that is, a function where [more…]

### How to Solve (and Factor) a Quadratic Equation with the Quadratic Formula

A quadratic equation is any *second-degree* polynomial equation — that’s when the highest power of *x,*or whatever other variable is used, is 2. The solution or solutions of a quadratic equation, [more…]

### How to Write Riemann Sums with Sigma Notation

You can use sigma notation to write out the right-rectangle sum for a function. For example, say you’ve got *f*(*x*) = *x*^{2} + 1.

By the way, you don’t need sigma notation for the math that follows. It’s just [more…]

### How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Cosine

When you integrate a trig integral that includes cosine, and if the power of cosine is odd and positive, you can convert and then use substitution to integrate. To make this conversion, you need to know [more…]

### How to Integrate Tangent/Secant Problems with an Odd, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents and secants where the tangent power is odd and positive. You’ll need the tangent-secant version of the Pythagorean identity: [more…]

### How to Integrate Tangent/Secant Problems with an Even, Positive Power of Secant

Here’s how you integrate a trig integral that contains tangents and secants where the secant power is even and positive. Like with all tangent/secant integrals, you use the tangent-secant version of the [more…]

### How to Integrate Problems with an Even, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents (and no secant factors) where the tangent power is even and positive. [more…]