Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 02

{{#incat:MER QGQ flag|{{#incat:MER QGH flag|{{#incat:MER QGS flag|}}}}}}

MATH101 B April 2024

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9  •  Q10  •  Q11  •  Q12  •  Q13  •  Q14  •  Q15  •  Q16 (a)  •  Q16 (b)  •  Q17 (a)  •  Q17 (b)  •  Q17 (c)  •  Q17 (d)  •  Q18  •  Q19  •  Q20 (a)  •  Q20 (b)  •  Q20 (c)  •  

   Other MATH101 B Exams

  •  April 2024  •  

Question 02
Find the area of the region under the curve $y=8-2x^{2}$ and above the $x$-axis.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
It may be helpful to first visualize the curve by plotting $y=8-2x^{2}$ and identifying where it intersects the $x-$ axis. Next, you can think about what mathematical tools we can use to find the area under the curve between these points.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Intersection of the curve $y=8-2x^{2}$ with the $x-$axis. First, we identify the points where the curve intersects the $x-$ axis by solving $8-2x^{2}=0,$ which yields $x=\pm 2$. Thus, the region of interest lies above the x-axis on the interval $[-2,2]$. This is represented as the shaded region in the attached graph. To determine the enclosed area, we set up a definite integral to compute the area under the curve from $x=2$ to $x=2$: $$\int _{-2}^{2}(8-2x^{2})dx=\left[8x-{\frac {2x^{3}}{3}}\right]_{-2}^{2}={\frac {64}{3}}.$$

{{#incat:MER CT flag||

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Area between two curves, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

}}