Create a grid to provide upper and lower bounds on the area under the curve y=4x-x^2 using 8 rectangles of equal width for each bound

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Let's start by finding the x-intercepts.

`4x-x^2=0=>x(4-x)=0=>x=0 or x=4` 

If we graph the function we will notice that the area under the curve and above the x-axis is between the two intercept. Hence our interval is [0,4]. We need to divide that into 8 equal parts, thus `Deltax=4/8=0.5`

Let's find the area by using f(0) for the height of the first rectangle, etc.

`A_1=0.5(f(0)+f(.5)+f(1)+f(1.5)+f(2)+f(2.5)+f(3)+f(3.5))=>`

`A_1=0.5(0+1.75+3+3.75+4+3.75+3+1.75)=>`

`A_1=0.5*21=10.5`

`A_2=0.5*(f(0.5)+f(1)+f(1.5)+f(2)+f(2.5)+f(3)+f(3.5)+f(4))=>`

`A_2=0.5(21+0)=10.5`

Given the symetric nature of the curve over the interval [0,4], we ended up with the lower and upper bound equal. 

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial