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### How to Approximate Area with Left Sums

You can approximate the area under a curve by using left sums. For example, say you want the exact area under a curve between two points, 0 and 3. The shaded area on the left graph in the below figure [more…]

### How to Approximate Area with Right Sums

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

### How to Approximate Area with Left Rectangles

You can approximate the area under a curve by summing up “left” rectangles. For example, say you want the area under the curve *f* (*x*) = *x*^{2} + 1 from 0 to 3. The shaded area of the graph on the left side [more…]

### 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 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 Area with the u-Substitution Method

You can use the Fundamental Theorem to calculate the area under a function (or just to do any old definite integral) that you integrate with the substitution method. What you want to do is to change the [more…]

### How to Determine the Dimensions for the Least Expensive Window Frame

You can use calculus to solve practical problems, such as determining the correct size for a home-improvement project. Here’s an example. A Norman window has the shape of a semicircle above a rectangle [more…]

### The Definition of the Definite Integral and How it Works

You can approximate the area under a curve by adding up right, left, or midpoint rectangles. To find an exact area, you need to use a definite integral. [more…]

### How the Area Function Works

The area function is a bit weird. Brace yourself. Say you’ve got any old function, *f*(*t*). Imagine that at some *t-*value, call it *s,* you draw a fixed vertical line. [more…]

### How to Find the Area between Two Curves

To find the area between two curves, you need to come up with an expression for a narrow rectangle that sits on one curve and goes up to another. [more…]

### How to Find the Area of a Surface of Revolution

A surface of revolution is a three-dimensional surface with circular cross sections, like a vase or a bell or a wine bottle. For these problems, you divide the surface into narrow circular bands, figure [more…]

### Find Exact Areas Under a Curve Using the Definite Integral

When approximating the area under a curve using left, right, or midpoint rectangles, the more rectangles you use, the better the approximation. So, "all [more…]