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
Citrus2011 [14]
3 years ago
15

Consider two functions f and g on [1, 8] such that integral^8_1 f(x) dx = 9, integral^8_1 g(x) dx = 5, integral^8_5 f(x) dx = 4,

and integral^5_1 g (x) dx = 3. Evaluate the following integrals. a. integral^5_1 2f(x) dx = (Simplify your answer.) b. integral^8_1 (f(x) - g (x)) dx = (Simplify your answer.) c. integral^5_1 (f (x) - g (x)) dx = (Simplify your answer.) d. integral^8_5 (g(x) - f(x)) dx = (Simplify your answer.) e. integral^8_5 7g(x) dx = (Simplify your answer.) f. integral^1_5 3f(x) dx = (Simplify your answer.)
Mathematics
1 answer:
Gelneren [198K]3 years ago
4 0

I'll abbreviate the definite integral with the notation,

I(f(x),a,b)=\int_a^bf(x)\,\mathrm dx

We're given

  • I(f,1,8)=9
  • I(g,1,8)=5
  • I(f,5,8)=4
  • I(g,1,5)=3

Recall that the definite integral is additive on the interval [a,b], meaning for some c\in[a,b] we have

I(f,a,b)=I(f,a,c)+I(f,c,b)

The definite integral is also linear in the sense that

I(kf+\ell g,a,b)=kIf(a,b)+\ell I(g,a,b)

for some constant scalars k,\ell.

Also, if a\ge b, then

I(f,a,b)=-I(f,b,a)

a. I(2f,1,5)=2I(f,1,5)=2(I(f,1,8)-I(f,5,8))=2(9-4)=\boxed{10}

b. I(f-g,1,8)=I(f,1,8)-I(g,1,8)=9-5=\boxed{4}

c. I(f-g,1,5)=I(f,1,5)-I(g,1,5)=\dfrac{I(2f,1,5)}2-I(g,1,5)=10-3=\boxed{7}

d. I(g-f,5,8)=I(g,5,8)-I(f,5,8)=(I(g,1,8)-I(g,1,5))-I(f,5,8)=(5-3)-4=\boxed{-2}

e. I(7g,5,8)=7I(g,5,8)=7(5-3)=\boxed{14}

f. I(3f,5,1)=3I(f,5,1)=-3I(f,1,5)=-\dfrac32I(2f,1,5)=-\dfrac32(10)=\boxed{-15}

You might be interested in
Any help would be great!
Virty [35]

Answer:

D

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Define Short Circuit.
ycow [4]
<span>Short Circuit- in a device, an electrical circuit of lower resistance than that of a normal circuit, typically resulting from the unintended contact of components and consequent accidental diversion of the current</span>
5 0
3 years ago
150 minutes per day = hours per week
RideAnS [48]
150 minutes / day for a week
= (150 minutes / day) * (7 days / week)
= 150 * 7 minutes / week
= 1050 minutes / week

(1050 minutes / week) * (1 hour / 60 minutes)
= (1050 / 60) hours / week
= (105 / 6) hours / week
= 17.5 hours / week

You could also convert to hours first. 150 minutes = 2.5 hours * 7 days = 17.5 hours per week.
4 0
3 years ago
Helppp plssss!!! Thanks!!
faltersainse [42]

The answer is D.

y = x + 1 = 4 + 1 = 5

y = 1/2x + 3 = 1/2(4) + 3 = 2 + 3 =5

4 0
3 years ago
A line passes through point A(12,18). A second point on the line has an x-value that is 125% of the x-value of point A and a y-v
Reil [10]

Answer:

(y - 18) = -3/2 (x - 12)

Step-by-step explanation:

125% of 12 equals 15

75% of 18 equals 13 1/2

so the slope equals 4.5/-3 or -3/2

7 0
2 years ago
Other questions:
  • What is the surface area of storage box measures 15 cm xby12 cm by 10 cm?
    6·1 answer
  • WILL GIVE A BRAINLIEST!!
    6·1 answer
  • Cuál es el punto medio entre el 0.1 y 0.2
    5·2 answers
  • Which graph shows the solution set of the compound inequality 1.5x-1&gt;6.5 or 7x+3&lt;-25
    5·1 answer
  • D+fg=h (solve for the variable “g”)
    14·1 answer
  • Find the locus of a point such that the sum of its distance from the point ( 0 , 2 ) and ( 0 , -2 ) is 6.
    15·2 answers
  • Please help! will give brainliest
    14·2 answers
  • Need the answer to part a and part b please :)
    11·1 answer
  • Solve 5x + 17 = 4(3x - 1) x = -3 x = -3 Infinitely many solutions Infinitely many solutions No solutions No solutions x = 3
    8·2 answers
  • Which is a way to use prime factorization to find the least common multiple of 8 and 6?
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!