Use the rectangles to approximate the area of the region. (Round your answer to three decimal places.)

Question

f(x) = 7/x

    [1, 5]

Give the exact area obtained using a definite integral.

Answer 1

integral of 7/x = -7lnx

evaluate from 1 to 5

7(ln5-ln1) = 7ln5 = 7(1.609437912) = 11.26606539 = 11.266 for 3 decimals

7/x = 7x^-1. Normal rule would be to increase exponent by one and divide by that number but that leads to division by 2. integral of 1/x = natural log of x

Use a billion rectangles you'll get virtually the same area

y=7/x is a hyperbola, the standard hyperbola rotated 45 degrees, 2 branches in quadrant I and III, midpoint between them is the origin.

do one rectangle with midpoint x=3., the midpoint between 1 and 5. f(3) = 7/3. Multiply xf(x) = 3(7/3) = 7. One rectangle is within almost 4 of the exact area. Using more rectangles will get closer, larger than 7, closer to 11.266.

with 2 rectangles the midpoints of the rectangles are 2 and 4. f(2)=7/2, f(4) = 7/4

2(7/2) = 7 + 4(7/4) = 14.

7 was within 4.266 of the exact answer

14 was within 2.734 of the exact answer

the more rectangles, the closer you get