1. Volume Between Two Curves Calculator

Learning Objectives. Determine the area of a region between two curves by integrating with respect to the independent variable. Find the area of a compound region. Determine the area of a region between two curves by integrating with respect to the dependent variable.In Introduction to Integration, we developed the concept of the definite integral to calculate the area below a curve on a given interval. In this section, we expand that idea to calculate the area of more complex regions. We start by finding the area between two curves that are functions of (displaystyle x), beginning with the simple case in which one function value is always greater than the other.

Volume Between Two Curves Calculator

We then look at cases when the graphs of the functions cross. Last, we consider how to calculate the area between two curves that are functions of (displaystyle y). Key Concepts. Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. To find the area between two curves defined by functions, integrate the difference of the functions. If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals and add the results to find the area of the region.

Sometimes it can be easier to integrate with respect to y to find the area. The principles are the same regardless of which variable is used as the variable of integration. The LibreTexts libraries are and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Unless otherwise noted, LibreTexts content is licensed. Have questions or comments?

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We have seen previously that for continuous functions defined on closed intervals, theFundamental Theorem of Calculus relates the process of finding antiderivatives tocalculating certain areas. As it turns out the process used to transcribe Riemannsums that approximate such areas to definite integrals that give an exactanswer is a fundamental procedure that applies to numerous examples in bothmathematics and other STEM fields. In the upcoming sections, we will apply thisprocess to many different cases of interest. As an opening remark, we dealonly with piecewise continuously differentiable functions unless otherwisenoted.The Fundamental Theorem of Calculus and AreasWe begin the section with an example that illustrates the concepts that arefundamental in setting up definite integrals. Consider a continuous function that ispositive on a closed interval in its domain. Suppose that we are interested in findingthe area bounded by the graph of and the -axis between.

Below is an exampleof such an example. We can write this area as the definite integraland we may use the Fundamental Theorem of Calculus to evaluate it. However,recalling how this result was obtained in the first place is instructive.

Understandingthe logic behind it is essential in order to apply a similar method to set up integralsto model other types of situations. We thus give a detailed conceptual outline of theargument here.Step 1: SliceSince we have expressed as an function of, we divide the interval into pieces ofuniform width. The servant leader ken blanchard pdf: full version software. As a note, using rectangles of equal width is not a requirement. In a more theoreticaltreatment, this can be treated, but it suffices for us to use uniform widths in order topresent more conceptually tractable examples.

As such, we adopt this conventionhere as well as in the coming sections.Step 2: ApproximateWe cannot determine the exact area of the slice, but we can approximate eachslice by a rectangle. The top of the rectangle should coincide with the topcurve, and the bottom of the rectangle should coincide with the lower curveabove some common -value on the base of the rectangle. For the sake ofthe picture, we use the lefthand endpoint to determine the height of eachrectangle. We can now find the area of the -th rectangle.where here is the -value of the right-hand endpoint of each rectangle. The height ofthe rectangle is thus.Let denote the total area obtained by adding the areas of the rectanglestogether.

Then, we can compute easily by adding up the areas of all of therectangles.or if you prefer using sigma notation,Note that as we use more rectangles, the following occur simultaneously:. The width of each rectangle decreases. The total number of rectangles increases. The sum of the areas of the rectangles becomes closer to the actual area.The actual area is then.Step 3: IntegrateWhile this infinite limit can be quite cumbersome to work out in even thesimplest cases, the Fundamental Theorem of Calculus comes to the rescue. Itguarantees that since is continuous on, we may find the area by findingantiderivatives and evaluating the difference of an antiderivative of at theendpoints.where is an antiderivative of.This can now be interpreted conceptually as follows:. The notation “” represents the finite but small width of a rectangle. Thenotation “” represents the infinitesimal width of a rectangle.

The procedure of definite integration can be thought of conceptuallyas follows. We simultaneously shrink the widths of the rectangles whileadding all of the areas together.

The integrand can be thought of as the area of an infinitesimal rectangleof height and thickness. As such, we cannot think of a rectangle of widthas having width zero since we must add infinitely many such rectanglestogether.This same procedure can be used to model many other situations, which will be thesubject of the coming sections. It is therefore important to understand the logicbehind it.The Area Between Two CurvesThe above procedure also can be used to find areas between two curves as well.Henceforth, by “area”, we will mean “total area”; the area bounded by the curvesshould be taken to be positive. For example, the area bounded by and from and isshown below. The area of the -th rectangle is given bywhere is the -value of the left-hand endpoint of the -th rectangle.To find the height of the darkly shaded rectangle, notice that this height isjust the vertical distance between the curves. Vertical distances can alwaysbe found by taking the top -value minus the bottom -value. Since these-values lie on the graphs of the given functions, at a given -value, we have and.Thus, the height of the rectangle is.The approximate total area obtained by adding the areas of the rectangles betweenand together.

As you can see, something interesting happens here; the curve used to determine theheight of the top rectangle changes. In order to express this area by integrating withrespect to, we have to split it into two pieces.The top curve will change at the -value where the two curves intersect. To find this-value, we first must express each curve as a function of. The function is already afunction of.

For the line, we can solve.We can now find the -value of the intersection point. Write with me:Hence, or.Note that by squaring both sides to eliminate the square root, we may haveintroduced an extraneous root. We can check this easily enough:By substituting into the equation, we obtain, which is a true statement.However, doing the same for gives, which is not true (though it should beclear why is a solution to the equation that results from squaring bothsides!)Thus, we use.For the second region, we find the rightmost -value is (set ).We can now think of the original region in two separate parts and write down anintegral that gives the area of each. The function used to determine the rightmost -value is and the function used to determine the leftmost -value is.The length of the rectangle is thus and the area is given by the integralEvaluating this (which you should work out yourself and confirm) givesChoosing a variable of integrationAs we have seen, choosing a particular type of slice may be more advantageous thananother. To make this more explicit, draw a vertical rectangle or horizontal rectanglein the region. If the top or bottom curve of the region depends on where you draw theslice, you’ll need more than one integral with respect to to find the area. If the left or right curve of the region depends on where you draw the slice,you’ll need more than one integral with respect to to find the area.

How many integrals with respect to are needed to compute this area?How many integrals with respect to are needed to compute this area?We conclude the section by summarizing a few important facts. To find vertical distances, we always take. To find horizontal distances,we always take. When we integrate with respect to, we use vertical slices and when weuse vertical slices, we integrate with respect to. When we integrate withrespect to, we use horizontal slices and when we use horizontal slices, weintegrate with respect to. Once we choose a variable of integration, everything in the integrand mustbe expressed in terms of that variable!

This includes both the limits ofintegration and any functions that arise in the integrand.