Part 1 of 5
Estimate the area under $f(x)=3x+10$ on $[0,3]$ using $n$ rectangles of equal width and the right side to measure the height.
area estimate $=$
(Enter an expression in terms of $n$ )

Step 1 ：We are asked to estimate the area under the curve $f(x)=3x+10$ on the interval $[0,3]$ using $n$ rectangles of equal width and the right side to measure the height.

Step 2 ：The width of each rectangle is the length of the interval divided by the number of rectangles, which is $\frac{3}{n}$.

Step 3 ：The height of each rectangle is the value of the function at the right endpoint of the subinterval. The right endpoint of the $i$-th subinterval is $\frac{3i}{n}$, so the height of the $i$-th rectangle is $f\left(\frac{3i}{n}\right)=3\left(\frac{3i}{n}\right)+10$.

Step 4 ：The area of the $i$-th rectangle is then the width times the height, which is $\frac{3}{n}(3\left(\frac{3i}{n}\right)+10)$.

Step 5 ：The total area is the sum of the areas of all the rectangles, which is $\sum _{i=1}^n\frac{3}{n}(3\left(\frac{3i}{n}\right)+10)$.

Step 6 ：Simplifying the expression, we get the total area under the curve $f(x)=3x+10$ on $[0,3]$ using $n$ rectangles of equal width and the right side to measure the height is given by the expression $30+\frac{27}{2}(\frac{n^2}{n^2}+\frac{n}{n^2})$.

Step 7 ：Final Answer: The area estimate is $\boxed{{\displaystyle 30+\frac{27}{2}(\frac{n^2}{n^2}+\frac{n}{n^2})}}$.