Question 7
Determine $\displaystyle\int_1^3(2x+3)\,dx$
Not quite. 2 is the width of the interval, $3-1$. The integral is an area. This region is 2 wide and between 5 and 9 tall, so its area is much more than 2. In 2021, 2.8% of Queensland students chose (A).
Not quite. 4 is $f(3)-f(1)=9-5$, the change in the integrand $2x+3$. Substitute the bounds into the antiderivative $x^2+3x$ instead. In 2021, 7.0% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2021, 82.1% of Queensland students chose (C).
Not quite. Check the antiderivative and the subtraction. $F(x)=x^2+3x$ gives $F(3)=18$ and $F(1)=4$. As a sense check, the region is a trapezium 2 wide with an average height of $\tfrac{5+9}{2}=7$. In 2021, 7.7% of Queensland students chose (D).
From 1 to 3 the area is 14, and both methods agree. The width, 2, and the rise, 4, are options (A) and (B), but neither one is the area. That’s option (C).
Very close, but not equal. The upper bound is , not 3. Use the button below.
With both bounds at 1 there is no width, so the integral is 0. Drag the upper bound to the right.
From 1 to the area is . The antiderivative and the rectangle plus triangle give the same number. The question’s upper bound is 3.
Antidifferentiate each term. Raise the power by one and divide by the new power.
$\displaystyle\int(2x+3)\,dx$$=x^2+3x+c$
For a definite integral you can leave out $c$. It cancels when you subtract.
Substitute the upper bound, then the lower bound, into the antiderivative.
$F(3)=9+9=18$$F(1)=1+3=4$
Subtract: upper minus lower.
$$18-4=14$$
Sense check: under a straight line the region is a trapezium. It is 2 wide with parallel sides 5 and 9, so its area is $\tfrac{5+9}{2}\times2=14$.
$$\begin{aligned}&\int_1^3(2x+3)\,dx\\&=\Bigl[x^2+3x\Bigr]_1^3\\&=(9+9)-(1+3)\\&=14\end{aligned}$$
The trapezium under the line agrees: width 2, heights 5 and 9, area 14.
How Queensland students answered in 2021
From the QCAA Mathematical Methods subject report for 2021. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2021 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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