Math / Calculus

Question(c) Estimate $\int_{0}^{4}\left(x^{2}+1\right) d x$ using a right-hand sum with $n=3$ (i.e. $\left.R_{3}\right)$ .
Round your answer to two decimal places. $R_{3}=$ $\square$

Studdy Solution

STEP 1

1. We are estimating the integral $\int_{0}^{4}(x^{2}+1) \, dx$ .
2. We are using a right-hand Riemann sum with $n=3$ .
3. The interval $[0, 4]$ is divided into 3 equal subintervals.

STEP 2

1. Determine the width of each subinterval.
2. Identify the right endpoints of each subinterval.
3. Evaluate the function at each right endpoint.
4. Calculate the right-hand sum $R_3$ .
5. Round the result to two decimal places.

STEP 3

Calculate the width of each subinterval $\Delta x$ :
$\Delta x = \frac{b-a}{n} = \frac{4-0}{3} = \frac{4}{3}$

STEP 4

Identify the right endpoints of each subinterval. Since $\Delta x = \frac{4}{3}$ , the right endpoints are:
$x_1 = \frac{4}{3}, \quad x_2 = \frac{8}{3}, \quad x_3 = 4$

STEP 5

Evaluate the function $f(x) = x^2 + 1$ at each right endpoint:
$f\left(\frac{4}{3}\right) = \left(\frac{4}{3}\right)^2 + 1 = \frac{16}{9} + 1 = \frac{25}{9}$
$f\left(\frac{8}{3}\right) = \left(\frac{8}{3}\right)^2 + 1 = \frac{64}{9} + 1 = \frac{73}{9}$
$f(4) = 4^2 + 1 = 16 + 1 = 17$

STEP 6

Calculate the right-hand sum $R_3$ :
$R_3 = \Delta x \left(f\left(\frac{4}{3}\right) + f\left(\frac{8}{3}\right) + f(4)\right)$
$R_3 = \frac{4}{3} \left(\frac{25}{9} + \frac{73}{9} + 17\right)$
Convert 17 to a fraction with denominator 9:
$17 = \frac{153}{9}$
$R_3 = \frac{4}{3} \left(\frac{25}{9} + \frac{73}{9} + \frac{153}{9}\right)$
$R_3 = \frac{4}{3} \left(\frac{251}{9}\right)$
$R_3 = \frac{4 \times 251}{27}$
$R_3 = \frac{1004}{27}$

STEP 7

Convert $\frac{1004}{27}$ to a decimal and round to two decimal places:
$R_3 \approx 37.19$
The estimated value of the integral using the right-hand sum $R_3$ is:
$\boxed{37.19}$

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Question(c) Estimate $\int_{0}^{4}\left(x^{2}+1\right) d x$ using a right-hand sum with $n=3$ (i.e. $\left.R_{3}\right)$ .
Round your answer to two decimal places. $R_{3}=$ $\square$

Studdy Solution

STEP 1

1. We are estimating the integral $\int_{0}^{4}(x^{2}+1) \, dx$ .
2. We are using a right-hand Riemann sum with $n=3$ .
3. The interval $[0, 4]$ is divided into 3 equal subintervals.

STEP 2

1. Determine the width of each subinterval.
2. Identify the right endpoints of each subinterval.
3. Evaluate the function at each right endpoint.
4. Calculate the right-hand sum $R_3$ .
5. Round the result to two decimal places.

STEP 3

Calculate the width of each subinterval $\Delta x$ :
$\Delta x = \frac{b-a}{n} = \frac{4-0}{3} = \frac{4}{3}$

STEP 4

Identify the right endpoints of each subinterval. Since $\Delta x = \frac{4}{3}$ , the right endpoints are:
$x_1 = \frac{4}{3}, \quad x_2 = \frac{8}{3}, \quad x_3 = 4$

STEP 5

Evaluate the function $f(x) = x^2 + 1$ at each right endpoint:
$f\left(\frac{4}{3}\right) = \left(\frac{4}{3}\right)^2 + 1 = \frac{16}{9} + 1 = \frac{25}{9}$
$f\left(\frac{8}{3}\right) = \left(\frac{8}{3}\right)^2 + 1 = \frac{64}{9} + 1 = \frac{73}{9}$
$f(4) = 4^2 + 1 = 16 + 1 = 17$

STEP 6

STEP 7

Convert $\frac{1004}{27}$ to a decimal and round to two decimal places:
$R_3 \approx 37.19$
The estimated value of the integral using the right-hand sum $R_3$ is:
$\boxed{37.19}$

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Question(c) Estimate ∫04(x2+1)dx\int_{0}^{4}\left(x^{2}+1\right) d x∫04​(x2+1)dx using a right-hand sum with n=3n=3n=3 (i.e. R3)\left.R_{3}\right)R3​).Round your answer to two decimal places. R3=R_{3}=R3​= □\square□

Studdy Solution

STEP 1

1. We are estimating the integral ∫04(x2+1) dx\int_{0}^{4}(x^{2}+1) \, dx∫04​(x2+1)dx.2. We are using a right-hand Riemann sum with n=3n=3n=3.3. The interval [0,4][0, 4][0,4] is divided into 3 equal subintervals.

STEP 2

1. Determine the width of each subinterval.2. Identify the right endpoints of each subinterval.3. Evaluate the function at each right endpoint.4. Calculate the right-hand sum R3R_3R3​.5. Round the result to two decimal places.

STEP 3

Calculate the width of each subinterval Δx\Delta xΔx:Δx=b−an=4−03=43\Delta x = \frac{b-a}{n} = \frac{4-0}{3} = \frac{4}{3}Δx=nb−a​=34−0​=34​

STEP 4

Identify the right endpoints of each subinterval. Since Δx=43\Delta x = \frac{4}{3}Δx=34​, the right endpoints are:x1=43,x2=83,x3=4x_1 = \frac{4}{3}, \quad x_2 = \frac{8}{3}, \quad x_3 = 4x1​=34​,x2​=38​,x3​=4

STEP 5

STEP 6

STEP 7

Convert 100427\frac{1004}{27}271004​ to a decimal and round to two decimal places:R3≈37.19R_3 \approx 37.19R3​≈37.19 The estimated value of the integral using the right-hand sum R3R_3R3​ is:37.19\boxed{37.19}37.19​

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Question(c) Estimate $\int_{0}^{4}\left(x^{2}+1\right) d x$ using a right-hand sum with $n=3$ (i.e. $\left.R_{3}\right)$ .
Round your answer to two decimal places. $R_{3}=$ $\square$

1. We are estimating the integral $\int_{0}^{4}(x^{2}+1) \, dx$ .
2. We are using a right-hand Riemann sum with $n=3$ .
3. The interval $[0, 4]$ is divided into 3 equal subintervals.

1. Determine the width of each subinterval.
2. Identify the right endpoints of each subinterval.
3. Evaluate the function at each right endpoint.
4. Calculate the right-hand sum $R_3$ .
5. Round the result to two decimal places.

Calculate the width of each subinterval $\Delta x$ :
$\Delta x = \frac{b-a}{n} = \frac{4-0}{3} = \frac{4}{3}$

Identify the right endpoints of each subinterval. Since $\Delta x = \frac{4}{3}$ , the right endpoints are:
$x_1 = \frac{4}{3}, \quad x_2 = \frac{8}{3}, \quad x_3 = 4$

Convert $\frac{1004}{27}$ to a decimal and round to two decimal places:
$R_3 \approx 37.19$
The estimated value of the integral using the right-hand sum $R_3$ is:
$\boxed{37.19}$