Area Between Curves We already know how to find the area between a curve and the x -axis. Next, we will progress to finding the area enclosed between two curves .

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Area Between Curves EXAMPLE 1: Find the area enclosed between the graphs of and between x = 1 and x = 4.

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Area Between Curves EXAMPLE 1: Find the area enclosed between the graphs of and between x = 1 and x = 4. Let’s start by looking at what we already know.

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Area Between Curves What definite integral represents this shaded area?

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Area Between Curves What definite integral represents this shaded area?

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Area Between Curves What definite integral represents this shaded area?

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Area Between Curves What definite integral represents this shaded area?

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Area Between Curves Let’s look at what we already know and what we are trying to find.

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Area Between Curves

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Area Between Curves

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Area Between Curves Pink Area - Beige Area = Green Area

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Area Between Curves Since both integrals have the same upper & lower limits of integration, we can combine them into a single integral.

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Area Between Curves Since both integrals have the same upper & lower limits of integration, we can combine them into a single integral.

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Area Between Curves Since both integrals have the same upper & lower limits of integration, we can combine them into a single integral.

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Area Between Curves I like to draw a “ representative rectangle ” in the region – as one rectangle of many in a Riemann sum.

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Area Between Curves I like to draw a “ representative rectangle ” in the region – as one rectangle of many in a Riemann sum. I look at the rectangle to see what function is the upper boundary and what function is the lower boundary .

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Area Between Curves I like to draw a “ representative rectangle ” in the region – as one rectangle of many in a Riemann sum. I look at the rectangle to see what function is the upper boundary and what function is the lower boundary .

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Area Between Curves Question : Does this method of “upper boundary – lower boundary” work all of the time? Does it work when part of the region is below the x-axis?

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Area Between Curves Question : Does this method of “upper boundary – lower boundary” work all of the time? Does it work when part of the region is below the x-axis? Let’s find the area of this shaded region, and let’s examine if the “upper – lower” method works.

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Area Between Curves What definite integrals represent areas of the two shaded regions, the yellow and the peach colored regions? Let’s begin by breaking the region into two smaller pieces .

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Area Between Curves Signed Area : The yellow region has positive area and the peach region has negative area .

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Area Between Curves (+) - (-) (+) + (+)

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Area Between Curves (+) - (-) (+) + (+)

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Area Between Curves

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Area Between Curves Let’s find the area of a region that is not bounded on the right and left by dashed lines. Can we use the “upper – lower” technique?

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Area Between Curves Let’s find the area of a region that is not bounded on the right and left by dashed lines. Can we use the “upper – lower” technique? Let’s draw a representative rectangle .

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Area Between Curves Can we use the “upper – lower” technique? Let’s draw a representative rectangle . Imagine the rectangle sliding both left and right with its height adapting to fit the region . Does the upper boundary remain the same? Does the lower boundary remain the same?

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Area Between Curves We can write a single definite integral. What are the lower & upper limits of integration?

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Area Between Curves We can write a single definite integral. Can we simplify the integrand before “integrating”?

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Area Between Curves We can write a single definite integral.

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Area Between Curves We can write a single definite integral.

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Area Between Curves In the problem we just did, we were given the graph and the points of intersection. For our last example, let’s try one in which we are given only the functions that enclose a region.

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Area Between Curves Find the area of the region enclosed by the graphs of and .

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Area Between Curves Find the area of the region enclosed by the graphs of and . What shapes are the two graphs?

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Area Between Curves Find the area of the region enclosed by the graphs of and . What shapes are the two graphs? p arabola opening down l ine sloping down left to right What must these graphs look like together if they enclose a region?

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Area Between Curves Find the area of the region enclosed by the graphs of and . This must be how the graphs meet. How can we find the points of intersection?

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Area Between Curves Find the area of the region enclosed by the graphs of and . How can we find the points of intersection?

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Area Between Curves Find the area of the region enclosed by the graphs of and . How can we find the points of intersection?

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Area Between Curves Find the area of the region enclosed by the graphs of and . How can we find the points of intersection?

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Area Between Curves Find the area of the region enclosed by the graphs of and . How can we find the points of intersection?

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Area Between Curves Find the area of the region enclosed by the graphs of and . Let’s draw the representative rectangle, then write our definite integral.

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Area Between Curves Find the area of the region enclosed by the graphs of and .

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Area Between Curves Find the area of the region enclosed by the graphs of and .

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Area Between Curves Find the area of the region enclosed by the graphs of and .

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Area Between Curves Find the area of the region enclosed by the graphs of and .

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Area Between Curves What have we learned about finding the area between curves? 1. 2. Representative rectangle 3. Find points of intersection. 4. Write a single definite integral, simplify the integrand, then evaluate.

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Area Between Curves That’s all for now, folks! Assignment: • Sect. 5.5: #1, 4, 7, 8, 10, 12 - 15, * 21 • Sect. 5.3: #2 • Document “ Introduction to Indefinite Integrals”

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