What is the net area between #f(x) = (x-2)^3 # and the x-axis over #x in [1, 5 ]#?

Answer 1

#20.5#

The integrand #f(x) = (x-2)^3# crosses 0 at #x=2#. The area enclosed between #x=1# and #x=2# is given by
#A_1 = |int_1^2 (x-2)^3dx|# #qquadquad = |(x-2)^4/4]_1^2| # #qquadqquad = |-1/4| = 1/4#
while, the area enclosed between #x=2# and #x=5# is given by
#A_2 = |int_2^5 (x-2)^3dx|# #qquadquad = |(x-2)^4/4]_2^5| # #qquadqquad = |81/4| = 81/4#
Thus, the net area is #81/4+1/4=82/4=20.5#
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 net area between ( f(x) = (x-2)^3 ) and the x-axis over ( x ) in ([1, 5]), you need to calculate the definite integral of ( f(x) ) from 1 to 5. The integral represents the area bounded by the function and the x-axis within the specified interval.

[ \text{Net area} = \int_{1}^{5} (x-2)^3 , dx ]

To find this integral, you can use the power rule for integration:

[ \int x^n , dx = \frac{x^{n+1}}{n+1} + C ]

Applying this rule to ( (x-2)^3 ), you'll get:

[ \int (x-2)^3 , dx = \frac{(x-2)^4}{4} + C ]

Now, evaluate this expression at the upper and lower bounds of the interval and subtract the lower bound value from the upper bound value:

[ \text{Net area} = \left[\frac{(5-2)^4}{4} - \frac{(1-2)^4}{4}\right] ]

[ = \left[\frac{3^4}{4} - \frac{(-1)^4}{4}\right] ]

[ = \left[\frac{81}{4} - \frac{1}{4}\right] ]

[ = \frac{80}{4} ]

[ = 20 ]

Therefore, the net area between ( f(x) = (x-2)^3 ) and the x-axis over ( x ) in ([1, 5]) is ( 20 ) square units.

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