Integral chasing
Can you find the missing constants from these not-quite-so-obvious definite integrals?
Age
16 to 18
Challenge level
Problem
Below is a selection of integrals, some of which you can't do directly and some that require the use of other mathematical skills.
Can you find the missing positive numbers $a$ to $d$?
$$\text{(1)} \ \int_a^5 10x+3 \ \text{d}x = 114$$
$$\text{(2)} \ \int_{2a}^9 b\sqrt{x}+\dfrac{a}{\sqrt{x}}\ \text{d}x=42$$
$$\text{(3)} \ \int_{\frac{1}{2}}^1 \dfrac{1}{x^5}-\dfrac{1}{x^2} \ \text{d}x=\dfrac{c+1}{4}$$
$$\text{(4)} \ \int^{c+2}_6 x^{\frac{b}{a}}\left(\sqrt{x}-\dfrac{1}{\sqrt{x}}\right) \ \text{d}x=ab^ad^a$$
This is an Underground Mathematics resource.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Getting Started
Once you have found the missing positive numbers $a$ to $d$, can you use them in the statement below to test your values?
The area formed between the $x$-axis and the lines $x=b$ and $x=d$, and the curve $y=(x-2a)(x+1)$ is $\dfrac{cd}{a(a+b)}$.
Remember, when we are looking at area, what must we check about the curve between the lines $x=b$ and $x=d$ when it is plotted?
The area formed between the $x$-axis and the lines $x=b$ and $x=d$, and the curve $y=(x-2a)(x+1)$ is $\dfrac{cd}{a(a+b)}$.
Remember, when we are looking at area, what must we check about the curve between the lines $x=b$ and $x=d$ when it is plotted?
Student Solutions
Hayley from Kimberley STEM College has managed to find the values of a,b,c and d which are:
$a=2$
$b=3$
$c=10$
$d=5$
Here is her working.