Use trapezoids with 6 equal subdivisions to approximate the area beneath the curve y=f(x)=1/(1+x^2) on the interval [0,6].

Question

Answers can be 1.406, 1.791, 0.896, 1.411, or 2.822

Answer 1

6 equal divisions of the domain [0,6] gives us [0,1], [1,2], [2,3], [3,4], [4,5], [5,6].

f(0)=1, f(1)=1/2, f(2)=1/5, f(3)=1/10, f(4)=1/17, f(5)=1/26, f(6)=1/37.

The trapezoids each consist of a rectangle and a triangle.

The area of the rectangles is f(x+1) because the width is 1.

The area of the triangles is (1/2)(f(x)-f(x+1)).

So the sum of the areas of the trapezoids is

(1/2+(1/2)(1-1/2))+(1/5+(1/2)(1/2-1/5))+

(1/10+(1/2)(1/5-1/10))+(1/17+(1/2)(1/10-1/17))+

(1/26+(1/2)(1/17-1/26))+(1/37+(1/2)(1/26-1/37))=

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

1/2+1/5+1/10+1/17+1/26+1/37+(1/2)(1-1/37)=1.4108 approx or 1.411 answer option 4.