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6.1 Areas Between Curves 1 Dr. Erickson
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6.1 Areas Between Curves2 How can we find the area between these two curves? We could split the area into several sections, use subtraction and figure it out, but there is an easier way. Dr. Erickson
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6.1 Areas Between Curves3 Consider a very thin vertical strip. The length of the strip is: or Since the width of the strip is a very small change in x, we could call it dx. Dr. Erickson
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6.1 Areas Between Curves4 Since the strip is a long thin rectangle, the area of the strip is: If we add all the strips, we get: Dr. Erickson
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6.1 Areas Between Curves5Dr. Erickson
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The area of the region bounded by the curves y = f(x) and y = g(x) and the lines x = a and x = b where f and g are continuous and f(x) g(x) for all x in [a, b] is given by 6.1 Areas Between Curves6 y = f(x) y = g(x) Dr. Erickson
![The area of the region bounded by the curves y = f(x) and y = g(x) and the lines x = a and x = b where f and g are continuous and f(x) g(x) for all x in [a, b] is given by 6.1 Areas Between Curves6 y = f(x) y = g(x) Dr.](http://images.slideplayer.com./26/8455277/slides/slide_6.jpg "Erickson.")
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6.1 Areas Between Curves7 If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y. Dr. Erickson
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6.1 Areas Between Curves8 We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y. length of strip width of strip Dr. Erickson
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6.1 Areas Between Curves9 1 Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.) Sketch the curves. 2 3 Write an expression for the area of the strip. If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y. 4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.) 5 Integrate to find area. Dr. Erickson
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Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Find the area of the region. 6.1 Areas Between Curves10Dr. Erickson
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Evaluate the integral and interpret it as the area of a region. Sketch the region. 6.1 Areas Between Curves11Dr. Erickson
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Use a graph to find the x-coordinates of the points of intersection of the given curves. Then find (possibly approximately) the area of the region bounded by the curves. 6.1 Areas Between Curves12Dr. Erickson
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