121616949-math.156

# 121616949-math.156 - Figure 7.1 Approximating the area...

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142 Chapter 7 Integration how to compute areas of rectangles, so we approximate the area by rectangles. Jumping straight to the general case, suppose we divide the interval between 0 and x into n equal subintervals, and use a rectangle above each subinterval to approximate the area under the curve. There are many ways we might do this, but let’s use the height of the curve at the left endpoint of the subinterval as the height of the rectangle, as in figure 7.1 . The height of rectangle number i is then 3( i - 1)( x/n ), the width is x/n , and the area is 3( i - 1)( x 2 /n 2 ). The total area of the rectangles is (0) x n + 3(1) x 2 n 2 + 3(2) x 2 n 2 + 3(3) x 2 n 2 + · · · + 3( n - 1) x 2 n 2 . By factoring out 3 x 2 /n 2 this simplifies to 3 x 2 n 2 (0 + 1 + 2 + · · · + ( n - 1)) = 3 x 2 n 2 n 2
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Unformatted text preview: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 7.1 Approximating the area under y = 3 t with rectangles. What you will have noticed, of course, is that while the problem in the second example appears to be much diﬀerent than the problem in the ﬁrst example, and while the easy approach to problem one does not appear to apply to problem two, the “approximation” approach works in both, and moreover the calculations are identical. As we will see, there are many, many problems that appear much diﬀerent on the surface but that turn out to...
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