How do you find the area between #f(x)=-x^2+4x+1, g(x)=x+1#?

Answer 1

Please see the explanation.

Find the boundaries of the area by setting #f(x) = g(x)#

#-x^2 + 4x + 1 = x + 1#

#-x^2 + 3x = 0#

#x = 0 and x = 3#

This is confirmed by the graph of the two functions:

Let the lower limit of integration# = 0#
Let the upper limit of integration# = 3#

The area between the two fuctions is:

#int_0^3 f(x)dx - int_0^3 g(x)dx =#

#int_0^3 (-x^2 + 4x + 1)dx - int_0^3 (x + 1)dx =#

#int_0^3 (-x^2 + 4x + 1) - (x + 1)dx =#

#int_0^3 (-x^2 + 3x)dx = (-x^3/3 + (3x^2)/2)|_0^3 = 4.5#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the area between the curves ( f(x) = -x^2 + 4x + 1 ) and ( g(x) = x + 1 ), first, determine their intersection points by setting ( f(x) = g(x) ) and solving for ( x ). Then, integrate the absolute difference between the two functions over the interval between these intersection points. The area can be calculated as ( \int_{a}^{b} |f(x) - g(x)| , dx ), where ( a ) and ( b ) are the x-coordinates of the intersection points.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7