Find the value of f on the top "9" on the bottom "1" x^2dx using four rectangles of equal width, with each left end-point used to find the height of each rectangle.
a) 2(1+9+25+49)
b) 2(9+25+49+81)
c) 0.5(1+9+25+49)
d) 0.5(9+25+49+81)
each rectangle has width (9-1)/4 = 2
using left endpoints, that would mean you want
2(f(1)+f(3)+f(5)+f(7)) = 2(1+9+25+49)
To find the value of the integral ∫(x^2)dx using four rectangles of equal width and using the left endpoint of each rectangle to determine its height, you will need to follow these steps:
1. Divide the interval [1, 9] into four equal subintervals. Since there are four rectangles, each rectangle will have a width of (9 - 1) / 4 = 8 / 4 = 2.
2. Determine the left endpoints of each subinterval. The left endpoints can be found by starting at the left endpoint of the whole interval (which is 1 in this case) and then adding the width of each subinterval successively. In this case, the left endpoints would be 1, 3, 5, and 7.
3. Evaluate the function f(x) = x^2 at each of the left endpoints to find the height of each rectangle. Calculate f(1), f(3), f(5), and f(7), which gives you 1^2, 3^2, 5^2, and 7^2, respectively.
4. Multiply each height by the width of the rectangles to find the area of each rectangle. In this case, since the width is 2 for each rectangle, you can multiply each height by 2.
5. Add up the areas of the four rectangles to get the approximate value of the integral. In this case, you need to add the areas of the four rectangles, which are obtained by multiplying each height by the width and then summing them up. Let's call this sum S.
Now, let's calculate the value of each option and see which one matches the result:
a) 2(1 + 9 + 25 + 49)
= 2(84)
= 168
b) 2(9 + 25 + 49 + 81)
= 2(164)
= 328
c) 0.5(1 + 9 + 25 + 49)
= 0.5(84)
= 42
d) 0.5(9 + 25 + 49 + 81)
= 0.5(164)
= 82
Based on the calculations, the correct answer is option c) 0.5(1 + 9 + 25 + 49), which gives the value 42.