b) Average value of $f(x)=x^{2}$ on $[5,13]$ :

View step-by-step help

Home

Math Problems

Precalculus

Operations with rational exponents

Full solution

Q. b) Average value of

f(x)=x^{2}

[5,13]

Formula Application: To find the average value of a function on an interval $[a, b]$ , use the formula: Average value = $\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx$ .
Denominator Calculation: First, let's plug in our function and limits into the formula: Average value = $(1/(13-5)) \times \int_{5}^{13} x^2 \,dx$ .
Antiderivative Calculation: Now, calculate the denominator: $13 - 5 = 8$ .
Antiderivative Evaluation: So, the formula becomes: Average value = $(\frac{1}{8}) \times \int_{5}^{13} x^2 \, dx$ .
Subtraction Calculation: Next, find the antiderivative of $x^2$ , which is $(1/3)x^3$ .
Final Multiplication: Now, evaluate the antiderivative from $5$ to $13$ : $\left[\left(\frac{1}{3}\right) * 13^3\right] - \left[\left(\frac{1}{3}\right) * 5^3\right]$ .
Final Multiplication: Now, evaluate the antiderivative from $5$ to $13$ : $\left[\frac{1}{3} \times 13^3\right] - \left[\frac{1}{3} \times 5^3\right]$ .Calculate each part: $\frac{1}{3} \times 13^3 = \frac{1}{3} \times 2197 = 732.333\ldots$ and $\frac{1}{3} \times 5^3 = \frac{1}{3} \times 125 = 41.666\ldots$ .
Final Multiplication: Now, evaluate the antiderivative from $5$ to $13$ : $\left[\frac{1}{3} \times 13^3\right] - \left[\frac{1}{3} \times 5^3\right]$ .Calculate each part: $\frac{1}{3} \times 13^3 = \frac{1}{3} \times 2197 = 732.333\ldots$ and $\frac{1}{3} \times 5^3 = \frac{1}{3} \times 125 = 41.666\ldots$ .Subtract the two values: $732.333\ldots - 41.666\ldots = 690.666\ldots$ .
Final Multiplication: Now, evaluate the antiderivative from $5$ to $13$ : $\left[\frac{1}{3} \times 13^3\right] - \left[\frac{1}{3} \times 5^3\right]$ .Calculate each part: $\frac{1}{3} \times 13^3 = \frac{1}{3} \times 2197 = 732.333\ldots$ and $\frac{1}{3} \times 5^3 = \frac{1}{3} \times 125 = 41.666\ldots$ .Subtract the two values: $732.333\ldots - 41.666\ldots = 690.666\ldots$ .Finally, multiply by the reciprocal of the denominator: $\frac{1}{8} \times 690.666\ldots = 86.333\ldots$ .