Problem

Find $\frac{d y}{d r}$ for $y=\int_{0}^{r} \sqrt{11+12 t^{2}} d t$

Solution

Step 1 :Given the function \(y=\int_{0}^{r} \sqrt{11+12 t^{2}} d t\), we are asked to find \(\frac{d y}{d r}\).

Step 2 :This is a problem of differentiation under the integral sign, also known as Leibniz's rule.

Step 3 :The derivative of the integral of a function from a constant to a variable with respect to that variable is simply the function evaluated at that variable.

Step 4 :So, we need to find the derivative of the integral of \(\sqrt{11+12t^2}\) from 0 to r with respect to r, which is simply \(\sqrt{11+12r^2}\).

Step 5 :Using the formula for the derivative of an integral, we find that \(\frac{d y}{d r} = \sqrt{11+12r^2}\).

Step 6 :However, this can be simplified further.

Step 7 :By applying the rules of differentiation and simplification, we find that \(\frac{d y}{d r} = \frac{6r^{2}}{\sqrt{12r^{2} + 11}} + \frac{\sqrt{12r^{2} + 11}}{2} + \frac{\sqrt{11}}{2\sqrt{\frac{12r^{2}}{11} + 1}}\).

Step 8 :Final Answer: \(\boxed{\frac{6r^{2}}{\sqrt{12r^{2} + 11}} + \frac{\sqrt{12r^{2} + 11}}{2} + \frac{\sqrt{11}}{2\sqrt{\frac{12r^{2}}{11} + 1}}}\)

From Solvely APP
Source: https://solvelyapp.com/problems/8005/

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