Math / Calculus

Question[-/1 Points] DETAILS MY NOTES
Evaluate the integral. $\int_{7}^{9}\left(x^{2}+2 x-7\right) d x$ $\square$

Studdy Solution

STEP 1

1. The integral is a definite integral from $x = 7$ to $x = 9$ .
2. The function to integrate is a polynomial: $x^2 + 2x - 7$ .
3. We will use the Fundamental Theorem of Calculus to evaluate the integral.

STEP 2

1. Find the antiderivative of the function $x^2 + 2x - 7$ .
2. Evaluate the antiderivative at the upper limit of integration.
3. Evaluate the antiderivative at the lower limit of integration.
4. Subtract the value found in step 3 from the value found in step 2 to find the definite integral.

STEP 3

Find the antiderivative of the function $x^2 + 2x - 7$ .
The antiderivative of $x^2$ is $\frac{x^3}{3}$ . The antiderivative of $2x$ is $x^2$ . The antiderivative of $-7$ is $-7x$ .
Thus, the antiderivative of the function is:
$F(x) = \frac{x^3}{3} + x^2 - 7x$

STEP 4

Evaluate the antiderivative at the upper limit of integration, $x = 9$ .
$F(9) = \frac{9^3}{3} + 9^2 - 7 \times 9$
$F(9) = \frac{729}{3} + 81 - 63$
$F(9) = 243 + 81 - 63$
$F(9) = 261$

STEP 5

Evaluate the antiderivative at the lower limit of integration, $x = 7$ .
$F(7) = \frac{7^3}{3} + 7^2 - 7 \times 7$
$F(7) = \frac{343}{3} + 49 - 49$
$F(7) = \frac{343}{3}$

STEP 6

Subtract the value found in STEP_3 from the value found in STEP_2 to find the definite integral.
$\int_{7}^{9} (x^2 + 2x - 7) \, dx = F(9) - F(7)$
$= 261 - \frac{343}{3}$
Convert $261$ to a fraction with a denominator of 3:
$261 = \frac{783}{3}$
$\int_{7}^{9} (x^2 + 2x - 7) \, dx = \frac{783}{3} - \frac{343}{3}$
$= \frac{783 - 343}{3}$
$= \frac{440}{3}$
The value of the definite integral is:
$\boxed{\frac{440}{3}}$

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Question[-/1 Points] DETAILS MY NOTES
Evaluate the integral. $\int_{7}^{9}\left(x^{2}+2 x-7\right) d x$ $\square$

Studdy Solution

STEP 1

1. The integral is a definite integral from $x = 7$ to $x = 9$ .
2. The function to integrate is a polynomial: $x^2 + 2x - 7$ .
3. We will use the Fundamental Theorem of Calculus to evaluate the integral.

STEP 2

1. Find the antiderivative of the function $x^2 + 2x - 7$ .
2. Evaluate the antiderivative at the upper limit of integration.
3. Evaluate the antiderivative at the lower limit of integration.
4. Subtract the value found in step 3 from the value found in step 2 to find the definite integral.

STEP 3

Find the antiderivative of the function $x^2 + 2x - 7$ .
The antiderivative of $x^2$ is $\frac{x^3}{3}$ . The antiderivative of $2x$ is $x^2$ . The antiderivative of $-7$ is $-7x$ .
Thus, the antiderivative of the function is:
$F(x) = \frac{x^3}{3} + x^2 - 7x$

STEP 4

Evaluate the antiderivative at the upper limit of integration, $x = 9$ .
$F(9) = \frac{9^3}{3} + 9^2 - 7 \times 9$
$F(9) = \frac{729}{3} + 81 - 63$
$F(9) = 243 + 81 - 63$
$F(9) = 261$

STEP 5

Evaluate the antiderivative at the lower limit of integration, $x = 7$ .
$F(7) = \frac{7^3}{3} + 7^2 - 7 \times 7$
$F(7) = \frac{343}{3} + 49 - 49$
$F(7) = \frac{343}{3}$

STEP 6

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Question[-/1 Points] DETAILS MY NOTESEvaluate the integral. ∫79(x2+2x−7)dx\int_{7}^{9}\left(x^{2}+2 x-7\right) d x∫79​(x2+2x−7)dx □\square□

Studdy Solution

STEP 1

1. The integral is a definite integral from x=7 x = 7 x=7 to x=9 x = 9 x=9.2. The function to integrate is a polynomial: x2+2x−7 x^2 + 2x - 7 x2+2x−7.3. We will use the Fundamental Theorem of Calculus to evaluate the integral.

STEP 2

STEP 3

STEP 4

STEP 5

Evaluate the antiderivative at the lower limit of integration, x=7 x = 7 x=7.F(7)=733+72−7×7 F(7) = \frac{7^3}{3} + 7^2 - 7 \times 7 F(7)=373​+72−7×7F(7)=3433+49−49 F(7) = \frac{343}{3} + 49 - 49 F(7)=3343​+49−49F(7)=3433 F(7) = \frac{343}{3} F(7)=3343​

STEP 6

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Question[-/1 Points] DETAILS MY NOTES
Evaluate the integral. $\int_{7}^{9}\left(x^{2}+2 x-7\right) d x$ $\square$

1. The integral is a definite integral from $x = 7$ to $x = 9$ .
2. The function to integrate is a polynomial: $x^2 + 2x - 7$ .
3. We will use the Fundamental Theorem of Calculus to evaluate the integral.

Evaluate the antiderivative at the lower limit of integration, $x = 7$ .
$F(7) = \frac{7^3}{3} + 7^2 - 7 \times 7$
$F(7) = \frac{343}{3} + 49 - 49$
$F(7) = \frac{343}{3}$