AP Calc AB Review for QZ 27-29

What does this picture represent?

• Left Riemann Sum

• Right Riemann Sum

• Midpoint Riemann Sum

• Trapezoidal Sum

Answer explanation

The picture represents a Left Riemann Sum, which approximates the area under a curve by using the left endpoints of subintervals to form rectangles. This method sums the heights of these rectangles to estimate the integral.

Tags

Left Riemann Sum

What does this picture represent?

• Left Riemann Sum

• Right Riemann Sum

• Midpoint Riemann Sum

• Trapezoidal Sum

Answer explanation

The picture represents a Right Riemann Sum, where the height of each rectangle is determined by the function value at the right endpoint of each subinterval, effectively approximating the area under the curve.

Tags

Right Riemann Sum

What does picture represent?

• Left Riemann Sum

• Right Riemann Sum

• Midpoint Riemann Sum

• Trapezoidal Sum

Answer explanation

The picture represents a Midpoint Riemann Sum, which approximates the area under a curve by using the function values at the midpoints of subintervals. This method often provides a more accurate estimate than left or right sums.

Tags

Midpoint Riemann Sum

What is the approximate area under the curve, using 4 intervals and left Riemann sum?

• 20

• 14

• 10

• 8

Answer explanation

To find the area using the left Riemann sum with 4 intervals, calculate the height of the function at the left endpoints and multiply by the width of each interval. This results in an approximate area of 10, making it the correct choice.

Tags

Left Riemann Sum

Based on the table, use a left Riemann sum and 4 sub-intervals to estimate the area under the curve. (Choose the correct set-up.)

• 5(3) + 1(4) + 2(5) + 1(7)

• 5(4) + 1(5) + 2(7) + 1(6)

• 5(3) + 6(4) + 8(5) + 9(7)

• 0(3) + 5(4) + 6(5) + 8(7)

Answer explanation

The left Riemann sum uses the left endpoints of the intervals. The correct choice, 5(3) + 1(4) + 2(5) + 1(7), corresponds to the heights at the left endpoints of the 4 sub-intervals, accurately estimating the area.

Tags

Left Riemann Sum

Based on the table, use a Right Riemann sum and 4 sub-intervals to estimate the area under the curve. (Choose the correct set-up.)

• 5(3) + 1(4) + 2(5) + 1(7)

• 5(4) + 1(5) + 2(7) + 1(6)

• 5(3) + 6(4) + 8(5) + 9(7)

• 0(3) + 5(4) + 6(5) + 8(7)

Answer explanation

To estimate the area using a Right Riemann sum with 4 sub-intervals, we take the function values at the right endpoints: 5(4) + 1(5) + 2(7) + 1(6), which corresponds to the correct choice.

Tags

Right Riemann Sum

Answer explanation

To evaluate \( \int_3^{11} f(x) \ dx \), use the property of integrals: \( \int_3^{11} f(x) \ dx = \int_3^1 f(x) \ dx + \int_1^{11} f(x) \ dx = -5 + 24 = 19 \). Thus, the answer is \( 19 \).

Tags

Accumulation Function

Estimate the area bounded by f(x) and the x-axis on [-5,5] using the trapezoidal method with 4 trapezoids.

• 159

• 152.5

• 79.5

• 76.25

Answer explanation

Using the trapezoidal method with 4 trapezoids on the interval [-5, 5], we calculate the area under f(x) by averaging the heights at the endpoints and multiplying by the width. This results in an estimated area of 79.5.

Tags

Trapezoidal Sum

• 58

• 30

• 29

• 28

Answer explanation

To find the left Riemann sum, divide the interval [3, 7] into 4 equal parts: [3, 4), [4, 5), [5, 6), [6, 7). The left endpoints are 3, 4, 5, and 6. Calculate f(3), f(4), f(5), f(6) and sum them, then multiply by the width (1). This gives 29.

Tags

Left Riemann Sum

• 17

• 7

• -17

• -7

Answer explanation

To find \( \int_7^{10} f(x) \ dx \), use the property of definite integrals: \( \int_0^{10} f(x) \ dx = \int_0^7 f(x) \ dx + \int_7^{10} f(x) \ dx \). Thus, \( 12 = 5 + \int_7^{10} f(x) \ dx \), leading to \( \int_7^{10} f(x) \ dx = 7 \).

Tags

Accumulation Function