Question 1, 5.1.2 Part 3 of 4 Estimate the area under the graph of $f(x)=5 x^{3}$ between $x=0$ and $x=2$ using each finite approximation below. a. A lower sum with two rectangles of equal width b. A lower sum with four rectangles of equal width c. An upper sum with two rectangles of equal width d. An upper sum with four rectangles of equal width a. The estimated area using a lower sum with two rectangles of equal width is 5 square units. (Type an integer or a decimal. Simplify your answer.) b. The estimated area using a lower sum with four rectangles of equal width is 11.25 square units. (Type an integer or a decimal. Simplify your answer.) c. The estimated area using an upper sum with two rectangles of equal width is square units (Type an integer or a decimal. Simplify your answer.)
Solution
Step 1 ：We are asked to estimate the area under the curve of the function \(f(x)=5x^3\) between \(x=0\) and \(x=2\) using an upper sum with two rectangles of equal width.
Step 2 ：The upper sum is calculated by taking the maximum value of the function on each subinterval as the height of the rectangle. Since we are using two rectangles of equal width, the width of each rectangle will be \((2-0)/2 = 1\). The subintervals are then \([0,1]\) and \([1,2]\).
Step 3 ：The maximum value of \(f(x)\) on the interval \([0,1]\) is \(f(1)=5(1)^3=5\) and the maximum value of \(f(x)\) on the interval \([1,2]\) is \(f(2)=5(2)^3=40\).
Step 4 ：The area of the first rectangle is then \(1*5=5\) and the area of the second rectangle is \(1*40=40\).
Step 5 ：The total area under the curve is then the sum of the areas of these two rectangles.
Step 6 ：Final Answer: The estimated area using an upper sum with two rectangles of equal width is \(\boxed{45}\) square units.
