Answer to Question #127349 in Calculus for Jack F

Question

Estimate the area under the graph of ƒ(x)=√(x+1) from x=−1 to x=3 using four rectangles and

- right endpoints,
- left endpoints,
- midpoints.

Answer 1

left endpoints -1 , 0, 1, 2,

f(-1) = (-1+1)1/2=0.00

f(0) = (0+1)1/2 = 1

f(1) = (1+1)1/2 = 1.41

f(2) = (2+1)1/2 = 1.73

S = 0 +1+1.41+1.73=4.14

right endpoints 0, 1, 2, 3

f(0) = (0+1)1/2=1

f(1) = (1+1)1/2 = 1.41

f(2) = (2+1)1/2 = 1.73

f(3) = (3+1)1/2 = 2

S = 1 +1.41+1.73+2 =6.14

 midpoints -0.5, 0.5, 1.5, 2.5

f(-0.5) = (-0.5+1)1/2=0.71

f(0.5) = (0.5+1)1/2 = 1.22

f(1.5) = (1.5+1)1/2 = 1.48

f(2.5) = (2.5+1)1/2 = 1.87

S = 0.71 +1.22+1.48+1.87=5.28

exact value of the integral f(x) F(x)= 2/3(x+1)3/2=5.33