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Section 7.1 – Area of a Region Between Two Curves

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White Board Challenge The circle below is inscribed into a square: What is the shaded area? 20 cm Calculator

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White Board Challenge Find the area of the region bounded by the function below and the x -axis between x = 1 to x = 6 : Calculator

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Area Between Two Curves The area of a region that is bounded above by one curve, y = f(x), and below by another y = g(x). The area is always POSITIVE.

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White Board Challenge Find the area of the region between y = sec 2 x and y = sin x from x = 0 to x = π/4 : TOPBOTTOM Between Outside In this example, all of the area was above the x-axis. Does the same process work for “negative” area? Calculator Subtracting the bottom area from the top, leaves only the area in-between. TOP BOTTOM

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Area Between Two Curves: Positive and Negative Area Find the area of the region between the two curves from x = a to x = b : Between (Positive) Between (Negative) TOPBOTTOM In this example, one area was positive and one was negative. Does the same process work if both areas are negative? THE SAME! TOPBOTTOM Subtracting the negative area switches it to adding a positive version. Must be positive!

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Area Between Two Curves: Negative Area Only Find the area of the region between the two curves from x = a to x = b : Outside Between (Negative) TOPBOTTOM In this example, both areas were negative. Now we can apply the three scenarios to any two curves. (Counted Twice) THE SAME! TOPBOTTOM Subtracting the negative area switches it to adding a positive version AND cancels the outside area.

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Area Between Two Curves: A Mix Find the area of the region between the two curves from x = a to x = b : TOPBOTTOM POS-POS POS-NEG NEG-NEG TOPBOTTOM

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Area Between Two Curves If f and g are continuous functions on the interval [a,b], and if f(x) ≥ g(x) for all x in [a,b], then the area of the region bounded above by y = f(x), below by y = g(x), on the left by x = a, and on the right by x = b is: TOPBOTTOM

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Reminder: Riemann Sums Recall that the integral is a limit of Riemann Sums:

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Example 1 Find the area of the region between the graphs of the functions Sketch a Graph Make Generic “Riemann” Rectangle(s) Base = dx Height = f – g Integrate the Area of Each Generic Rectangle Find the Boundaries/Intersections

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Example 2 Find the area of the region enclosed by the parabolas y = x 2 and y = 2x – x 2. Sketch a Graph Make Generic “Riemann” Rectangle(s) Base = dx Height = ( 2x–x 2 )–( x 2 ) Integrate the Area of Each Generic Rectangle Find the Boundaries/Intersections

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Example 3 Find the area of the region bounded by the graphs y = 8/x 2, y = 8x, and y = x. Sketch a Graph Make Generic “Riemann” Rectangle(s) Base = dx Height = 8x-x Integrate the Area of Each Generic Rectangle Find the Boundaries/Intersections Base = dx Height = 8/x 2 -x

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Example 4 Find the area of the region bounded by the curves y = sin x, y = cos x, x = 0, and x = π/2. Sketch a Graph Make Generic “Riemann” Rectangle(s) Base = dx Height = cos-sin Integrate the Area of Each Generic Rectangle Find the Boundaries/Intersections Base = dx Height = sin-cos What other Integrals could be used? (Symmetrical) (Keeps it Positive)

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Area Between Two Curves If f and g are continuous functions on the interval [a,b], then the area of the region bounded by y = f(x), y = g(x), on the left by x = a, and on the right by x = b is: It does not matter which function is greater. NOTE: There have been AP problems in the past that ask for an integral without an absolute value. So the first method is still important.

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No Calculator “Warm-up”: 1985 Section I NOW WE CAN DO!

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White Board Challenge Find the area enclosed by the line y = x – 1 and the parabola y 2 = 2x + 6. Calculator

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Example 5 Find the area enclosed by the line y = x – 1 and the parabola y 2 = 2x + 6. Sketch a Graph Make Generic “Riemann” Rectangle(s) Base = dy Height= ( y+1 )–( 1/2y 2 –3) Integrate the Area of Each Generic Rectangle Find the Boundaries/Intersections Sometimes Solve for x

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White Board Challenge Using two methods (one with dx and one with dy ), find the area between the x -axis and the two curves:

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