Question

Which of the following functions has the highest area under the curve between \(x=0\) and \(x=10\)?

• A. \(y=x+8\)
• B. \(y=3x\)
• C. \(y=2x+1\)
• D. \(y=x+2\)

Hint:

Compute the definite integral (or the trapezoid area) of each line from x=0 to x=10 and compare the four values.

Answer 1

The Correct Option is B

Since every option is a straight line and stays above the x-axis for $x$ between $0$ and $10$, the area under the curve is just the area of a trapezoid with parallel sides equal to the line's value at $x=0$ and at $x=10$, and width $10$. Use the trapezoid area formula $\text{Area} = \frac{1}{2}(y(0)+y(10)) \times 10$ instead of integrating.

1. $y=x+8$: $y(0)=8$, $y(10)=18$. Area $= \frac{1}{2}(8+18)(10) = \frac{1}{2}(26)(10) = 130$.
2. $y=3x$: $y(0)=0$, $y(10)=30$. Area $= \frac{1}{2}(0+30)(10) = \frac{1}{2}(30)(10) = 150$.
3. $y=2x+1$: $y(0)=1$, $y(10)=21$. Area $= \frac{1}{2}(1+21)(10) = \frac{1}{2}(22)(10) = 110$.
4. $y=x+2$: $y(0)=2$, $y(10)=12$. Area $= \frac{1}{2}(2+12)(10) = \frac{1}{2}(14)(10) = 70$.

Lining up all four trapezoid areas, $130$, $150$, $110$, $70$, the steepest line with the largest end value, $y=3x$, wins, since it climbs to $30$ at $x=10$ while starting from $0$, giving the widest average height overall.

Let's summarize:

• For a straight line staying positive on an interval, the area under it is a trapezoid area, found from the endpoint heights without doing integration.
• Comparing all four trapezoids shows $y=3x$ has the biggest area, $150$, over $x=0$ to $x=10$.

So the function with the highest area under the curve is $y=3x$, option (B).