Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Problem 1088

The area enclosed by the curves \(x^{2}=y, y=x+2\) and \(x\) -axis is...... (a) \((3 / 2)\) (b) \((5 / 2)\) (c) \((5 / 6)\) (d) \((7 / 6)\)

Short Answer

Expert verified
The area enclosed by the curves \(x^2 = y, y = x + 2\) and the x-axis is (b) \(\frac{5}{2}\).

Step by step solution

01

Find the Intersection Points

Let's first find the x-coordinates of the points where the two curves intersect. To do that, we set the two equations equal to each other: \(x^2 = x + 2\) Now, we can rearrange the equation to make it a quadratic equation: \(x^2 - x - 2 = 0\) Now, let's solve the quadratic equation for x.
02

Solve the Quadratic Equation

Using the quadratic formula, we can solve the equation \(x^2 - x - 2 = 0\). The quadratic formula is given by: \(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\) In our case, a = 1, b = -1, and c = -2. So, \(x = \frac{-( -1)\pm\sqrt{( -1)^2-4(1)( -2)}}{2(1)}\) Calculating, we get x = -1 and x = 2 as the x-coordinates of the intersection points.
03

Find the y-coordinates of Intersection Points

Now, let's find the y-coordinates of the intersection points by substituting the x-coordinates into either of our given equations (let's use the first one: x^2 = y): For x = -1: \(y = (-1)^2 = 1\) For x = 2: \(y = (2)^2 = 4\) So, the intersection points are (-1, 1) and (2, 4).
04

Set up the Area Integral

Now we can set up the integral, representing the area enclosed by the curves, between -1 and 2: \(A = \int_{-1}^2 [(x+2) - x^2] dx\)
05

Evaluate the Integral

Now, let's evaluate the integral: \(A = \int_{-1}^2 (x+2 - x^2) dx\) \(A = \left[\frac{x^2}{2} + 2x - \frac{x^3}{3}\right]_{-1}^2\) Now, let's find the value of the expression at the bounds and subtract: \((\frac{(2)^2}{2}+2(2)-\frac{(2)^3}{3}) - (\frac{(-1)^2}{2}+2(-1)-\frac{(-1)^3}{3})\) After evaluating the expression, we get: \(A = \frac{5}{2}\) So, the answer is (b) \(\frac{5}{2}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The area enclosed between the curves \(y=\log _{e}(x+e)\) and the coordinate axes is...... (a) 1 (b) 4 (c) 2 (d) 3

\(\quad 12-1012:(\pi / 9) \int_{0}(\tan x+\tan 2 x+\tan 3 x+\tan x \cdot \tan 2 x \cdot \tan 3 x) d x\) is equal to....... (a) \((1 / 3) \log 2\) (b) \(\log ^{3} \sqrt{4}\) (c) \(3 \log 2\) (d) \(4 \log \sqrt{3}\)

If \((\pi / 2) \int_{(\pi / 4)}\left[\left(\cos ^{n} x\right) /\left(\sin ^{n+2} x\right)\right] d x=[1 /(K-1)]\) then \(K\) is equal to \(\ldots \ldots .\) (a) \(n\) (b) \(n+1\) (c) \(n+2\) (d) \(n+3\)

The value of \((\pi / 4)\\}_{0}\left[\left(8 \tan ^{2} x+8 \tan x+8\right)\right.\) \(\left./\left(\tan ^{2} x+2 \tan x+1\right)\right] d x\) is \(\ldots \ldots\) (a) 0 (b) \(\pi\) (c) \(\pi+2\) (d) \(\pi-2\)

The value of the integral \({ }^{1} \int_{0}\left(x^{5}+6 x^{4}+5 x^{3}+4 x^{2}+3 x+1\right) e^{x-1} d x\) is equal to...... (a) 5 (b) \(5 \mathrm{e}\) (c) \(5 \mathrm{e}^{2}\) (d) \(5 \mathrm{e}^{4}\)
See all solutions

Recommended explanations on English Textbooks

Cues and Conventions

Read Explanation

Synthesis Essay

Read Explanation

History of English Language

Read Explanation

Essay Prompts

Read Explanation

Listening and Speaking

Read Explanation

Creative Story

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free