The left Riemann sum approximation for $$\int$$ₐᵇ f(x) dx using 10 equal subintervals is 45.2, and the right Riemann sum approximation is 52.8. Which of the following is the best approximation for the value of the integral?
Computing right Riemann sums from tables, formulas, and graphs.
23 questions
AP Learning Target(s): LT 2: Approximating Areas with Riemann Sums, LT 3: Riemann Sums, Summation Notation, and Definite Integral Notation
The left Riemann sum approximation for $$\int$$ₐᵇ f(x) dx using 10 equal subintervals is 45.2, and the right Riemann sum approximation is 52.8. Which of the following is the best approximation for the value of the integral?
49.0
98.0
52.8
45.2
For the function $$f(x) = x^3 - 2x$$ on the interval $$[0, 2]$$, which of the following statements is true about the left and right Riemann sum approximations of $$\int_{0}^{2} f(x) dx$$ using $$n = 4$$ subintervals of equal width?
The right Riemann sum is greater than the left Riemann sum
The trapezoidal sum equals the average of midpoint and right sums
The left Riemann sum overestimates the true value
The midpoint Riemann sum provides an exact value of the integral
Using the graph of $$f(x)=\sqrt{x}$$ on the interval $$[1,9]$$, estimate the definite integral $$\int_{1}^{9} \sqrt{x} dx$$ using a right Riemann sum with 4 equal subintervals.
Approximately $$21.50$$ if a left Riemann sum is mistakenly applied.
$$20.00$$
$$19.23$$
$$18.00$$
Let $$\displaystyle f$$ be the function defined by $$\displaystyle f(x) = 9^x$$. What is the right Riemann sum approximation of $$\displaystyle \int_{0}^{2} f(x) dx$$ using four subintervals of equal length?
$$\displaystyle 40$$
$$\displaystyle 20$$
$$\displaystyle 60$$
$$\displaystyle 80$$
What is the approximation of $$\int_{1}^{4} (x^3 - 2x^2 + 3) dx$$ using a right Riemann sum with three subintervals of equal width?
50
22
37
32
Let $$f$$ be the function defined by $$f(x) = x^3 - 2x^2 + 3$$. What is the value of the right Riemann sum with four subintervals of equal width used to approximate $$\int_{0}^{2} f(x) dx$$?
3.5
4.75
5
4
The approximation of $$\int_{0}^{5} x^2 dx$$ obtained from a right Riemann sum using five equal subintervals is
41.25
42.5
55
30
Refer to the same graph of $$f(x)=x^2-2*x+3$$ on the interval $$[0,5]$$. Using a right Riemann sum with 5 equal subintervals, approximate the area under the curve.
$$25$$
$$42$$
The incorrect use of left endpoints or averaging may lead one to an area of about $$35$$.
$$40$$
Which of the following is represented by the expression $$\lim_{n \to \infty} \sum_{i=1}^{n} f\left(a + i \frac{b-a}{n}\right) \frac{b-a}{n}$$?
$\displaystyle\int_{a}^{b}$ f(x) dx using midpoints
$\displaystyle\int_{a}^{b}$ f(x) dx using left endpoints
The average value of f on [a,b]
$\displaystyle\int_{a}^{b}$ f(x) dx using right endpoints
Which of the following expressions represents the right Riemann sum approximation for `f(x) = x² + 3x` on the interval `[1, 5]` using four equal subintervals?
1(2² + 3·2) + 1(3² + 3·3) + 1(4² + 3·4) + 1(5² + 3·5)
1(1.5² + 3·1.5) + 1(2.5² + 3·2.5) + 1(3.5² + 3·3.5) + 1(4.5² + 3·4.5)
1(1² + 3·1) + 1(2² + 3·2) + 1(3² + 3·3) + 1(4² + 3·4) + 1(5² + 3·5)
1(1² + 3·1) + 1(2² + 3·2) + 1(3² + 3·3) + 1(4² + 3·4)
The velocity of a particle moving along a line is described by the function $$v(t) = t^2 - 4t + 3$$ meters per second. Using a right Riemann sum with four subintervals of equal length, which of the following best approximates the total distance traveled by the particle on the interval $$[1, 5]$$?
24 meters
28 meters
12 meters
32.5 meters
Which of the following expressions represents a right Riemann sum?
∑[f(x_i)Δy]
∑[f(c_i)Δy]
∑[f(c_i)Δx]
∑[f(x_i)Δx]
The table gives selected values for a function $$f$$. Which of the following statements is true about the right Riemann sum approximation of $$\int_{0}^{3} f(x) dx$$ using the three subintervals indicated by the table?
| $$x$$ | 0 | 1 | 2 | 3 |
| $$f(x)$$ | 2 | 3 | 5 | 7 |
The right Riemann sum equals $$12$$ by incorrectly averaging adjacent values.
The right Riemann sum equals $$10$$ by mistakenly using the left endpoints.
The right Riemann sum equals $$18$$ due to a miscalculation in subinterval widths.
The right Riemann sum equals $$15$$.
The table above shows selected values for a function $$f$$. Which of the following statements correctly describes the right Riemann sum approximation of $$\int_{0}^{4} f(x) dx$$ using two subintervals of equal width?
| x | 0 | 2 | 4 |
| f(x) | 1 | 3 | 7 |
The right Riemann sum equals $$20$$.
The right Riemann sum equals $$22$$ due to a miscalculation of the subinterval widths.
The right Riemann sum equals $$18$$ by incorrectly averaging the endpoints.
The right Riemann sum equals $$16$$ by mistakenly using left endpoints.
The table below provides selected values for a function `f`. Which of the following expressions correctly calculates the right Riemann sum for `f` on the interval `[1, 8]` using the three subintervals given in the table?
| t | 1 | 3 | 5 | 8 |
|---|---|---|---|---|
| f(t) | 2 | 5 | 9 | 14 |
Right Sum = $$ (3-1)*(5) + (5-3)*(9) + (8-5)*(14) = 70 $$
Right Sum = $$ (3-1)*(5) + (5-3)*(9) + (8-5)*(14+2) = 74 $$
Right Sum = $$ 2*(5+9+14) = 56 $$
Right Sum = $$ (3-1)*(2) + (5-3)*(5) + (8-5)*(9) = 58 $$
The table provides selected values for a function `f`. Approximate the area under the curve of `f` on the interval `[1, 6]` using a right Riemann sum with the subintervals given by the table.
| x | 1 | 2 | 4 | 6 |
|---|---|---|---|---|
| f(x) | 3 | 5 | 9 | 12 |
$$47$$
$$50$$
$$52$$
$$44$$
People leave a building during an evacuation drill at a rate modeled by the function $$R(t)$$, measured in people per minute, where $$t$$ is the time in minutes since the drill began. Selected values of $$R(t)$$ are shown in the table above. Using a right Riemann sum with three subintervals indicated by the table, what is the approximate total number of people who left the building during the first 15 minutes?
| $$t$$ (minutes) | 0 | 5 | 10 | 15 |
|---|---|---|---|---|
| $$R(t)$$ (people per minute) | 100 | 100 | 75 | 55 |
230
1375
1150
2075
A tank contains 50 liters of oil at time $$t = 4$$ hours. Oil is pumped into the tank at a rate $$R(t)$$, in liters per hour, where $$t$$ is measured in hours. Selected values of $$R(t)$$ are given in the table.
| $$t$$ (hours) | 4 | 7 | 12 | 15 |
|---|---|---|---|---|
| $$R(t)$$ (liters/hour) | 6.5 | 6.2 | 5.9 | 5.6 |
Using a right Riemann sum with three subintervals indicated by the data in the table, what is the approximate number of liters of oil in the tank at time $$t = 15$$ hours?
68.2
64.9
116.6
114.9
The function $$R$$ models the rate, in people per minute, at which people leave a building $$t$$ minutes after an evacuation drill begins. Selected values of $$R(t)$$ are shown in the table.
| $$t$$ (minutes) | 0 | 5 | 10 | 15 |
|---|---|---|---|---|
| $$R(t)$$ (people per minute) | 100 | 100 | 75 | 55 |
Using a right Riemann sum with three subintervals indicated by the table, what is the approximation for the total number of people who leave the building during the first 15 minutes?
230
2075
1150
1375
Which of the following expressions represents the definite integral $\displaystyle\int_{1}^{3}$ (2x³ - 5x) dx as the limit of a Riemann sum with right endpoints?
lim(n→∞) Σᵏ⁼ⁿₖ₌₁ (2(1+2(k-1)/n)³-5(1+2(k-1)/n))·(2/n)
lim(n→∞) Σᵏ⁼ⁿₖ₌₁ (2(1+2k/n)³-5(1+2k/n))·(2/n)
lim(n→∞) Σᵏ⁼ⁿₖ₌₀ (2(1+2k/n)³-5(1+2k/n))·(2/n)
lim(n→∞) Σᵏ⁼ⁿₖ₌₁ (2(1+2k/n)³-5(1+2k/n))·(1/n)
In a right Riemann sum, the height of each rectangle is determined by the value of the function at which of the following?
By using the average of the heights for each base
By using the midpoint of each base
By using the value of f(x) for the left side of each rectangle
By using the value of f(x) for the right side of each rectangle
Using the partition x₀ = 1, x₁ = 2, x₂ = 4, and x₃ = 5 for the interval [1, 5], which expression represents the right Riemann sum approximation for $\displaystyle\int_{1}^{5}$ ln(x) dx?
ln(1)(2-1) + ln(2)(4-2) + ln(4)(5-4)
ln(2) + 2ln(4) + ln(5) - ln(1)
ln(2)(2-1) + ln(4)(4-2) + ln(5)(5-4)
ln(2) + 2ln(4) + ln(5)
The table shows selected values for a function $$f$$. Approximate the area under the graph of $$f$$ on the interval $$[0, 9]$$ using a right-hand Riemann sum with the subintervals given in the table.
| x | f(x) |
|---|---|
| 0 | 3 |
| 4 | 7 |
| 6 | 10 |
| 9 | 15 |
56
87
93
101