The table shows selected values for a function f. What is the difference between the right and left Riemann sum approximations for the area under the graph of f, using the subintervals determined by the table?
| x | 0 | 1 | 3 | 6 |
|---|---|---|---|---|
| f(x) | 4 | 5 | 8 | 12 |
Computing left Riemann sums from tables, formulas, and graphs.
21 questions
AP Learning Target(s): LT 2: Approximating Areas with Riemann Sums, LT 3: Riemann Sums, Summation Notation, and Definite Integral Notation
The table shows selected values for a function f. What is the difference between the right and left Riemann sum approximations for the area under the graph of f, using the subintervals determined by the table?
| x | 0 | 1 | 3 | 6 |
|---|---|---|---|---|
| f(x) | 4 | 5 | 8 | 12 |
$$19$$
$$15$$
$$17$$
$$21$$
Let E(n) be the error resulting from approximating $\displaystyle\int_{0}^{2}$ e^x dx with a left Riemann sum using n equal subintervals. Which of the following statements about E(n) is true?
E(n) < 0 and lim(n→∞) E(n) = 0
E(n) alternates sign as n increases
E(n) > 0 and lim(n→∞) E(n) = e²-1
E(n) > 0 and lim(n→∞) E(n) = 0
The integral $\displaystyle\int_{1}^{4}$ √x dx is approximated by a left Riemann sum with 5 equal subintervals, denoted L₅. Which of the following is true?
L₅ < $\displaystyle\int_{1}^{4}$ √x dx
The relationship cannot be determined
L₅ = $\displaystyle\int_{1}^{4}$ √x dx
L₅ > $\displaystyle\int_{1}^{4}$ √x dx
The table below shows values for the function $$f(x) = x^2$$. What is the left Riemann sum approximation of the area under the graph of $$f$$ on the interval $$[1, 3]$$ using four equal subintervals?
| x | $$f(x)=x^2$$ |
|---|---|
| 1 | $$1$$ |
| 1.5 | $$2.25$$ |
| 2 | $$4$$ |
| 2.5 | $$6.25$$ |
| 3 | $$9$$ |
$$10.75$$
$$6.75$$
$$0.5*(1 + 2.25 + 4 + 6.25 + 9)$$
$$7.75$$
Consider the graph of the function $$f(x)=x^2-2*x+3$$ shown above. Using a left Riemann sum with 5 equal subintervals over the interval $$[0,5]$$, approximate the area under the curve (i.e. sum the areas of the rectangles).
$$24$$
$$27$$
$$25$$
Due to a misinterpretation of the rectangle widths and function values, the total area is approximately $$30$$.
Consider the function $$f(x) = \sqrt{(x+2)}$$ on the interval $$[0,8]$$. Using a left Riemann sum with 4 equal subintervals, which of the following expressions represents the exact approximate area under the curve?
$$6*\sqrt{2}+4+2*\sqrt{6}$$
$$4+2*\sqrt{6}+4*\sqrt{2}+2*\sqrt{10}$$
$$6*\sqrt{2}+4+4*\sqrt{6}$$
$$3*\sqrt{2}+2+\sqrt{6}$$
What is the left Riemann sum approximation of $$\int_{0}^{6} x^3 dx$$ using three subintervals of equal width?
144
306
576
360
Which type of Riemann sum approximates the area under a curve by using the left endpoint of each subinterval to determine the height of the rectangles?
Trapezoidal sum
Left Riemann sum
Right Riemann sum
Midpoint Riemann sum
The velocity of a particle, $v(t)$, at selected times $t$ is shown in the table above. Using a left Riemann sum with the three subintervals indicated by the table, what is the approximate displacement of the particle from $t=0$ to $t=6$?
| t (s) | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| v(t) (m/s) | 3 | 5 | 7 | 6 |
$$28$$
$$32$$
$$30$$
$$20$$
Which of the following expressions represents a left Riemann sum?
∑[f(c_i)Δx]
∑[f(x_i)Δy]
∑[f(x_i)Δx]
∑[f(c_i)Δy]
The table shows selected values for a function `f`.
| x | 1 | 2 | 4 | 5 |
|---|---|---|---|---|
| f(x) | 3 | 5 | 7 | 10 |
Consider the following expressions:
I. $$1(3) + 2(5) + 1(7) = 20$$
II. $$1(5) + 2(7) + 1(10) = 29$$
III. $$1(3) + 1(5) + 1(7) = 15$$
Which of these expressions represents a left Riemann sum approximation for `f` on the interval `[1, 5]` using the subintervals determined by the data in the table?
I only
III only
I and III
II only
The table provides selected values for a function `f`. Which of the following expressions represents a left Riemann sum approximation of $$\int_{0}^{7} f(x) dx$$ using the subintervals determined by the data in the table?
| x | 0 | 2 | 4 | 7 |
|---|---|---|---|---|
| f(x) | 1 | 6 | 10 | 15 |
Left Sum = $$ (2-0)*(1) + (4-2)*(6) + (7-4)*(15) = 59 $$
Left Sum = $$ 1*(1+6+10) = 17 $$
Left Sum = $$ (2-0)*(1) + (4-2)*(6) + (7-4)*(10) = 44 $$
Left Sum = $$ (2-0)*(2) + (4-2)*(4) + (7-4)*(7) = 33 $$
The table below provides selected values for a function $$f$$. Use a left Riemann sum with the subintervals indicated by the data in the table to approximate the value of $$\int_{1}^{8} f(x) dx$$.
| x | f(x) |
|---|---|
| 1 | 2 |
| 3 | 5 |
| 5 | 9 |
| 8 | 13 |
45
39
41
43
The table above shows values for a function $$f$$. Which of the following is true about the left Riemann sum approximation for the area under the graph of $$f$$ on the interval $$[0, 3]$$ using three subintervals of equal width?
| x | 0 | 1 | 2 | 3 |
| f(x) | 2 | 3 | 5 | 7 |
The left Riemann sum equals $$10$$.
The left Riemann sum equals $$15$$ because it incorrectly includes all function values.
The left Riemann sum equals $$12$$ by averaging only the first and last values.
The left Riemann sum equals $$14$$ by erroneously combining intermediate values.
The table below provides selected values for a function $f$. Use a left Riemann sum with the subintervals indicated by the data to approximate $$\int_{0}^{7} f(x) dx$$.
| x | 0 | 2 | 4 | 7 |
|---|---|---|---|---|
| f(x) | 1 | 6 | 10 | 15 |
$$40$$
$$44$$
$$42$$
$$46$$
The table provides selected values for a function $$g$$. Use a left Riemann sum with the subintervals indicated in the table to approximate $$\int_{0}^{6} g(x) dx$$.
| x | 0 | 1 | 3 | 6 |
| g(x) | 4 | 5 | 9 | 15 |
$$5*1+9*2+15*3$$
$$\frac{1}{2}(4+5)*1+\frac{1}{2}(5+9)*2+\frac{1}{2}(9+15)*3$$
$$4+5+9$$
$$41$$
The table provides selected values for a function `f`. Which of the following correctly computes the left Riemann sum approximation of $$\int_{0}^{4} f(x) dx$$ using the subintervals indicated by the table?
| x | 0 | 1 | 3 | 4 |
|---|---|---|---|---|
| f(x) | 4 | 5 | 7 | 10 |
Left Riemann Sum = $$1*(5)+2*(7)+1*(10)=29$$
Left Riemann Sum = $$\frac{1*(4+5)+2*(5+7)+1*(7+10)}{2}=25$$
Left Riemann Sum = $$\frac{21+29}{2}=25$$
Left Riemann Sum = $$1*(4)+2*(5)+1*(7)=21$$
Selected values of a continuous function $$f$$ are shown in the table below.
| $$x$$ | $$0$$ | $$25$$ | $$30$$ | $$50$$ |
|---|---|---|---|---|
| $$f(x)$$ | $$4$$ | $$6$$ | $$8$$ | $$12$$ |
What is the left Riemann sum approximation of $$\displaystyle\int_{0}^{50} f(x) dx$$ using the three subintervals indicated by the table?
380
290
360
390
The table shows selected values for a continuous function $$f$$. What is the left Riemann sum approximation of $$\displaystyle \int_{0}^{50} f(x) dx$$ using the subintervals $$[0, 25]$$, $$[25, 30]$$, and $$[30, 50]$$?
| $$x$$ | $$0$$ | $$25$$ | $$30$$ | $$50$$ |
|---|---|---|---|---|
| $$f(x)$$ | $$4$$ | $$6$$ | $$8$$ | $$12$$ |
290
390
380
360
Which of the following describes how the height of each rectangle is determined in a left Riemann sum?
By finding the value of f(x) for the right side of each rectangle.
By finding the value of f(x) for the left side of each rectangle.
By averaging the heights of the left and right sides of each rectangle.
By using the midpoint of each rectangle as the height.
All of the following statements about left Riemann sums are true EXCEPT:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| f(x) | 2 | 4 | 6 | 8 |
In a left Riemann sum, the function value used for each subinterval is taken from the right endpoint of that subinterval.
For a strictly increasing function, the left Riemann sum underestimates the actual value of the integral.
Increasing the number of subintervals generally improves the accuracy of the approximation.
A left Riemann sum approximates the integral by using the value of $$f(x)$$ at the left endpoint of each subinterval.