1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
melamori03 [73]
3 years ago
14

Let $f(x) = x^2$ and $g(x) = \sqrt{x}$. Find the area bounded by $f(x)$ and $g(x).$

Mathematics
1 answer:
Anna [14]3 years ago
6 0

Answer:

\large\boxed{1\dfrac{1}{3}\ u^2}

Step-by-step explanation:

Let's sketch graphs of functions f(x) and g(x) on one coordinate system (attachment).

Let's calculate the common points:

x^2=\sqrt{x}\qquad\text{square of both sides}\\\\(x^2)^2=\left(\sqrt{x}\right)^2\\\\x^4=x\qquad\text{subtract}\ x\ \text{from both sides}\\\\x^4-x=0\qquad\text{distribute}\\\\x(x^3-1)=0\iff x=0\ \vee\ x^3-1=0\\\\x^3-1=0\qquad\text{add 1 to both sides}\\\\x^3=1\to x=\sqrt[3]1\to x=1

The area to be calculated is the area in the interval [0, 1] bounded by the graph g(x) and the axis x minus the area bounded by the graph f(x) and the axis x.

We have integrals:

\int\limits_{0}^1(\sqrt{x})dx-\int\limits_{0}^1(x^2)dx=(*)\\\\\int(\sqrt{x})dx=\int\left(x^\frac{1}{2}\right)dx=\dfrac{2}{3}x^\frac{3}{2}=\dfrac{2x\sqrt{x}}{3}\\\\\int(x^2)dx=\dfrac{1}{3}x^3\\\\(*)=\left(\dfrac{2x\sqrt{x}}{2}\right]^1_0-\left(\dfrac{1}{3}x^3\right]^1_0=\dfrac{2(1)\sqrt{1}}{2}-\dfrac{2(0)\sqrt{0}}{2}-\left(\dfrac{1}{3}(1)^3-\dfrac{1}{3}(0)^3\right)\\\\=\dfrac{2(1)(1)}{2}-\dfrac{2(0)(0)}{2}-\dfrac{1}{3}(1)}+\dfrac{1}{3}(0)=2-0-\dfrac{1}{3}+0=1\dfrac{1}{3}

You might be interested in
5 Points |<br> Which of the following are remote interior angles of 6? Check all that apply.
aniked [119]

Answer: 1 and 3

Step-by-step explanation:

I had the same question on my test the other day hope this helped.

4 0
3 years ago
What is the general term equation, an, for the arithmetic sequence 5, 1, â3, â7, . . . , and what is the 21st term of this seque
Valentin [98]
Hello :

the nth term ( general term  of an arithmetic sequence is :

An =A1+(n-1)d   ....a common difference is : d , A1 : the first terme
A1 = 5    and : d = 1 - 5 = - 4    n = 21
A21 = 5 + (21 -1)(-4)
A21 = 5 - 80 = -75

<span>                                      
</span>

3 0
3 years ago
Help please please please
VLD [36.1K]

Answer:

10%

Step-by-step explanation:

10/100 = 1/10 = .1 = 10%

5 0
3 years ago
Read 2 more answers
Every seventh-grade student at Martin Middle School takes one world language class.
koban [17]

Answer:

s = 180

Step-by-step explanation:

s is the total number of seventh-grade students

1/10s take French which is 18

1/10s = 18

s = 18(10)

s = 180

6 0
2 years ago
2) A snail moving at 3.6 m/h; convert to mm/s
jolli1 [7]

Answer:

60mm/s

Step-by-step explanation:

Plz give brianliest and push the thanks button

6 0
3 years ago
Read 2 more answers
Other questions:
  • Claire's bedroom measures 16 feet by 18 feet if the purple carpet that she wants to buy costs $3.59 per square foot what will it
    10·1 answer
  • Yesterday, a Bakery baked 54 loaves of bread in 20 minutes today the bakery needs to bake 375 loaves of bread at this rate predi
    6·2 answers
  • Eli is following this recipe to bake bread rolls. He uses 700 ml of water. How much of the other ingredients does he use? Recipe
    6·1 answer
  • Can someone tell me how to do it
    11·2 answers
  • 95, 45, 37, 82, 90, 100, 91, 78,67,84,85,85, 82, 91,
    15·1 answer
  • How do you find the volume of compound shapes
    15·1 answer
  • I'm lost with this problem please help. Write answers as a fractions
    9·2 answers
  • Question 5
    7·1 answer
  • A measure of the amount of matter in a object
    6·1 answer
  • Need help pls I’m taking a test
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!