Definite intergrable

Question

Consider the definite integral ∫ 3 0 2 + √ 9 − x 2 dx.

(a) Approximate the integral with a LHS with n = 3.

(b) Approximate the integral with the Trapezoid Rule with n = 3.

(c) Approximate the integral with Simpson’s Rule with n = 6.

(d) Find the exact value of the integral using Geometry.

Answer 1

f(x) = 2 +√(9-x²)

We are finding the area from 0 to 3

a) n = 3. (3 -0)/3 = 1

LHS

f(0) + f(1) +f(2) =2 + √9 + 2 + √(9-1²) + 2 + √(9-2²) = 9 + 2√2 + √5 = 14.06

b) Trapezoidal Rule, f(3) = 2 + sqrt(9-3²) = 2

(f(0) + 2f(1)+2f(2)+f(3))/2 = (2 + 3 + 2(2 + √8) + 2(2 +√5) + 2)/2 = 15/2 + 2√2 + √5 = 12.56

c)

(f(0) + 4f(1/2)+2f(1)+4f(3/2)+2f(2)+4f(5/2)+f(3)/6 =

(2+3 + 4(2+1/2√35) + 2(2+√8) + 4(2+1/2√27) + 2(2 +√5) + 4(2+1/2√11) + 2)/6 = 13/2 + √35/3 + 2√2/3 + √3 + √5/3 + √11/3 = 13.00

d) The 2 term is a rectangle 2x3, which has area 6.

The √(9-x²) is a fourth of a circle with radius 3, which has area 1/4* πr² = 9/4π

The area is 6 + 9/4π = 13.07