Question

Consider the arc of the curve from point P to point Q: y = (1/2) * x^2, P(-5, 25/2), Q(5, 25/2). Set up an integral in terms of x that can be used to find the length of the arc. L = ∫(-5 to 5) [ dx. Find the length of the arc of the curve from point P to point Q.

Answer 1

To find the length of the arc of the curve from point P to point Q, use the arc length formula ∫(a to b) sqrt[1+(f'(x))^2] dx. Substitute the given curve y = (1/2) * x^2 and its derivative f'(x) = x into the formula. Evaluate the integral to find the length of the arc.

Step-by-step explanation:

To find the length of the arc of the curve from point P to point Q, we can use the arc length formula for a curve y=f(x):

L = ∫(a to b) sqrt[1+(f'(x)) ^2] dx

For the given curve y = (1/2) * x^2, the derivative is f'(x) = x. Substituting these values into the arc length formula:

L = ∫ (-5 to 5) sqrt[1+(x) ^2] dx

This integral can be evaluated to find the length of the arc.

Answer 2

The definite integral of the expression "x^2/12 + 2" from -5 to 5 is 485/18. This represents the total area under the curve defined by the expression between these limits.

Let's find the integral of the expression:

`$\int_(-5)^5\left((x^2)/(12)+2\right) d x$`

We'll use the power rule of integration to solve this problem. This rule states that the integral of
`$x^n$` is
`$x^{(n+1)/(n+1)$`, where n is any real number except for -1.

Steps to solve:

1. Simplify the expression:

`$\int\left((1)/(12) x^2+2\right) d x$`

2. Apply the power rule of integration:

`$\left[(1)/(12) \cdot (x^(2+1))/(2+1)+2 \cdot (x^(0+1))/(0+1)\right]_(-5)^5$`

3. Add the numbers:

`$\left[(1)/(12) \cdot (x^3)/(3)+2 \cdot (x^1)/(1)\right]_(-5)^5$`

4. Simplify:

`$\left[(1)/(36) x^3+2 x\right]_(-5)^5$`

5. Substitute and subtract:

`$(1)/(36) \cdot 5^3+2 \cdot 5-\left((1)/(36) \cdot(-5)^3+2(-5)\right)$`

6. Simplify:

`$(485)/(18)$`

The answer is
`$(485)/(18)$`.

The complete question is here: