Calculus Homework Question

Question

Approximate the area under the curve f(x)= 4x²+1 on the interval [2,5] using 4 rectangles of equal width. Get the heights from the right-hand sides.

Answer 1

Since we need 4 rectangles of equal width (the x distance), we need to take the difference between the starting and ending point of the given interval, and divide the result by 4.

 

(5 - 2) / 4 = 3 / 4 = 0.75

 

This means that the width of each rectangle is 0.75.   To get the height of each rectangle, we need to evaluate f(x) at each rectangle's right hand endpoint.

 

 

We know that f(x) is a positive parabola, so the height of the rectangles under this parabola will only increase.  It will also help to draw these rectangles in a coordinate system.  Keep in mind that the right hand sides of the rectangles are under the curve, so we need to evaluate f(x) at the rectangle right side points.

 

For the first rectangle in the interval [2, 2.75]:

 

f(2.75) = 4(2.75)² + 1

           = 31.25

 

 

For the second rectangle in the interval [2.75, 3.50]:

 

f(3.50) = 2(3.50)² + 1

           = 50

 

 

For the third rectangle in the interval [3.5, 4.25]:

 

f(4.25) = 4(4.25)² + 1

           = 73.25

 

 

For the fourth rectangle in the interval [4.25, 5]:

 

f(5) = 4(5)² + 1

           = 101

 

 

Now we add up all the areas of the rectangles.

 

 

Area under the curve = 0.75 * (31.25 + 50 + 73.25 + 101)

                                

                                = 191.625