Question

Consider the integral 8 (x2+1) dx 0 (a) Estimate the area under the curve using a left-hand sum with n = 4. 250 Is this sum an overestimate or an underestimate of the true value? overestimate underestimate (b) Estimate the area under the curve using a right-hand sum with n = 4. 248

Answer 1

(a) 120 square units (underestimate)

(b) 248 square units

Explanation:

(a) left sum

See the attachment for a diagram of the areas being summed (in orange). This is the sum of the first 4 table values for f(x), each multiplied by 2 (the width of the rectangle). Quite clearly, the curve is above the rectangle for the entire interval, so the rectangle area underestimates the area under the curve.

left sum = 2(1 + 5 + 17 + 37) = 2(60) = 120 . . . . square units

(b) right sum

The right sum is the sum of the last 4 table values for f(x), each multiplied by 2 (the width of the rectangle). This sum is ...

right sum = 2(5 +17 + 37 +65) = 2(124) = 248 . . . . square units