How do you find the area #y = f(x) = 4/x^2# , from 1 to 2 using ten approximating rectangles of equal widths and right endpoints?

Question

Charles Arocha · Accepted Answer

See the explanation.

#y = f(x) = 4/x^2# , on #[1,2]# using #n=10# approximating rectangles of equal widths and right endpoints. #Deltax = (b-a)/n = (2-1)/10=0.1# The bases of the rectangles are the subintervals: #[1,1.1], [1.1,1.2], [1.2,1.3], . . . , [1.8,1.9], [1.9,2]# The right endpoints are : #1.1, 1.2, 1.3, . . . , 1.9, 2# Use these to find the height of the rectangles: #f(1.1)=4/1.1^2, f(1.2)=4/1.2^2, . . . , f(1.9) = 4/1.9^2, f(2)=4/2^2# So the areas of the rectangles are #f(x_i)Deltax# for these various #x_i# and the Riemann sum is: #4/1.1^2Delta x+4/1.2^2 Delta x+ * * * +4/1.9^2Delta x+4/2^2Delta x# # = (4/1.1^2+4/1.2^2 + * * * +4/1.9^2+4/2^2)Delta x# # = (4/1.1^2+4/1.2^2 + * * * +4/1.9^2+4/2^2)(0.1)# Now do the arithmetic.

Christopher Agresta · Answer

undefined To find the area under the curve $ y = f(x) = \frac{4}{x^2} $ from $ x = 1 $ to $ x = 2 $ using ten approximating rectangles of equal widths and right endpoints, follow these steps:

1. Determine the width of each rectangle:
   $ \Delta x = \frac{2 - 1}{10} = 0.1 $.

2. Calculate the right endpoints for each rectangle:
   $ x_i = 1 + i \cdot \Delta x $, for $ i = 1, 2, ..., 10 $.

3. Evaluate the function $ f(x) $ at each right endpoint:
   $ y_i = f(x_i) = \frac{4}{(1 + i \cdot 0.1)^2} $, for $ i = 1, 2, ..., 10 $.

4. Calculate the area of each rectangle:
   $ A_i = \Delta x \cdot y_i $, for $ i = 1, 2, ..., 10 $.

5. Sum up the areas of all rectangles:
   $ A_{\text{total}} = A_1 + A_2 + ... + A_{10} $.

6. Compute the total area under the curve as the approximation:
   $ A_{\text{total}} = \sum_{i=1}^{10} \Delta x \cdot \frac{4}{(1 + i \cdot 0.1)^2} $.

Calculate the numerical value of $ A_{\text{total}} $ to find the area under the curve from $ x = 1 $ to $ x = 2 $ using ten rectangles with right endpoints.