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
Can someone write the point slope equation
almond37 [142]

okay so, first what is the slope of the line? you have stated it is -1.

so now we have a point on the line (x1,y1) = (4, -9), so now we can use:

y-y1 = m(x-x1)

y - (-9) = -1 (x - 4)

y + 9 = -x + 4

y = -x + 4 -9

y = -x -5

Let's check. If x = 4, then

y = - (4) -5 = -9

seems to be correct


6 0
3 years ago
Write the equation of a line with the slope, \frac{-1}{2}, which passes through the point (5, -6). write the answer in slope-int
Goryan [66]
Y= -1/2x +b
-6= -1/2(5) +b
-6 = -2.5 +b
-3.5 = b

Answer: y= -1/2x - 3.5
8 0
3 years ago
PLEASE HELP!!!! 20 POINTS!!!!
Otrada [13]
Your answer is c lmk if I’m wrong
7 0
2 years ago
If two distinct positive divisors of 64 are randomly selected, what is the probability that their sum will be less than 32?
scZoUnD [109]

Answer: The required probability is \dfrac{10}{21}

Step-by-step explanation:

Since we have given that

Divisors of 64= 1,2,4,8,16,32,64

Sum would be less than 32

Since it has 7 divisors in total.

If we select any two distinct divisors from first five divisors, the sum will always less than 32.

So, the probability that their sum will be less than 32 is given by

\dfrac{^5C_2}{^7C_2}\\\\=\dfrac{10}{21}

Hence, the required probability is \dfrac{10}{21}

8 0
3 years ago
PLEASE HELP
Sever21 [200]
Your answer should be y= -72
7 0
2 years ago
Other questions:
  • What is the volume of the composite figure below?
    13·2 answers
  • Scott baked a batch of rolls. He gave a bag of 10 rolls to each of 7 friends he kept 1 roll for himself. How many rolls did he b
    10·2 answers
  • QUICKKKKKKKKKKKKKKKKKKK PLEASE! I MIN ONLY!
    7·1 answer
  • A good walking pace would let you travel 4.5 miles in 90 minutes. What is your walking speed?
    14·1 answer
  • How do you solve this problem
    11·1 answer
  • Add: (x2 + 5x + 2) + (x2 - 2x + 7).
    10·1 answer
  • Brad swims 4 laps every 6min how long does it take him to swim 14 laps?
    8·1 answer
  • Match each equation to its graph. Use the drop-<br> down menus to describe the equations.
    9·1 answer
  • Simplify the expression:<br> 2 – 6p - 4 + -3
    8·1 answer
  • Consider the function.<br> f(x) = 2 x + 12<br> What is the y-intercept of f1(x)?
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!