Question

use finite approximations to estimate the area under the graph of the function f(x) = 3 - x² - 2x between x = -3 and x = 1 for each of the following cases.
a. using a lower sum with two rectangles of equal width
b. using a lower sum with four rectangles of equal width
c. using an upper sum with two rectangles of equal width
d. using an upper sum with four rectangles of equal width
a. the area under the graph of f(x) using a lower sum with two rectangles of equal width is 0. (type an integer or a decimal.)
b. the area under the graph of f(x) using a lower sum with four rectangles of equal width is 6. (type an integer or a decimal.)
c. the area under the graph of f(x) using an upper sum with two rectangles of equal width is 16. (type an integer or a decimal.)
d. the area under the graph of f(x) using an upper sum with four rectangles of equal width is \boxed{} (type an integer or a decimal.)

Explanation:

Step1: Find the width of each rectangle

The interval is from \( x = -3 \) to \( x = 1 \), so the length of the interval is \( 1 - (-3) = 4 \). We are using four rectangles, so the width \( \Delta x \) of each rectangle is \( \frac{4}{4} = 1 \). The subintervals are \([-3, -2]\), \([-2, -1]\), \([-1, 0]\), \([0, 1]\).

Step2: Determine the maximum value of \( f(x) \) on each subinterval (for upper sum)

• For \([-3, -2]\): \( f(x) = 3 - x^2 - 2x \). Let's find \( f(-3) = 3 - 9 + 6 = 0 \), \( f(-2) = 3 - 4 + 4 = 3 \). The maximum on this subinterval is \( 3 \) (at \( x = -2 \)).
• For \([-2, -1]\): \( f(-2) = 3 \), \( f(-1) = 3 - 1 + 2 = 4 \). The maximum is \( 4 \) (at \( x = -1 \)).
• For \([-1, 0]\): \( f(-1) = 4 \), \( f(0) = 3 - 0 - 0 = 3 \). The maximum is \( 4 \) (at \( x = -1 \)).
• For \([0, 1]\): \( f(0) = 3 \), \( f(1) = 3 - 1 - 2 = 0 \). The maximum is \( 3 \) (at \( x = 0 \)).

Step3: Calculate the upper sum

The upper sum is \( \sum_{i = 1}^{4} f(x_i^*) \Delta x \), where \( x_i^* \) is the point in each subinterval where \( f(x) \) is maximum. So:

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Answer:

14