assignment submission for this assignment, you submit answers by question parts. the number of submiss…

Answer:

$\frac{1}{36}-\frac{7}{36e^{6}}$

Explanation:

Step1: Use integration - by - parts formula

The integration - by - parts formula is $\int_{a}^{b}u\mathrm{d}v=uv|{a}^{b}-\int{a}^{b}v\mathrm{d}u$. Let $u = y$ and $\mathrm{d}v=e^{- 6y}\mathrm{d}y$. Then $\mathrm{d}u=\mathrm{d}y$ and $v=-\frac{1}{6}e^{-6y}$.

Step2: Apply the formula

$\int_{0}^{1}\frac{y}{e^{6y}}\mathrm{d}y=\left[- \frac{y}{6e^{6y}}\right]{0}^{1}+\frac{1}{6}\int{0}^{1}e^{-6y}\mathrm{d}y$.

Step3: Evaluate $\left[- \frac{y}{6e^{6y}}\right]_{0}^{1}$

$\left[- \frac{y}{6e^{6y}}\right]_{0}^{1}=-\frac{1}{6e^{6}}-0 =-\frac{1}{6e^{6}}$.

Step4: Evaluate $\frac{1}{6}\int_{0}^{1}e^{-6y}\mathrm{d}y$

First, find the antiderivative of $e^{-6y}$, which is $-\frac{1}{6}e^{-6y}$. Then $\frac{1}{6}\int_{0}^{1}e^{-6y}\mathrm{d}y=\frac{1}{6}\left[-\frac{1}{6}e^{-6y}\right]_{0}^{1}=\frac{1}{6}\left(-\frac{1}{6e^{6}}+\frac{1}{6}\right)=\frac{1}{36}-\frac{1}{36e^{6}}$.

Step5: Combine the results

$-\frac{1}{6e^{6}}+\frac{1}{36}-\frac{1}{36e^{6}}=\frac{1}{36}-\frac{7}{36e^{6}}$.