Part 1 of 5
Estimate the area under $f(x)=3 x+10$ 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 \(\frac{3}{n} \left(3\left(\frac{3i}{n}\right) + 10\right)\).

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

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 \(30 + \frac{27}{2} \left(\frac{n^2}{n^2} + \frac{n}{n^2}\right)\).

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