What is the area under #f(x)=5x-1# in #x in[0,2] #?

Answer 1

The net area is #8#, The actual area is #8.2 \ "unit"^2#

We seek the are under #f(x)=5x-1# where #x in [0,2]#

Method 1:

The bounded net area is that of a trapezium with heights:

# f(0) = -1 # # f(2) = 9 #
and width #2#, So we can use the trapezium formula:
# A=1/2(a+b)h # # \ \ \ =1/2(-1+9)(2) # # \ \ \ =8 #

Method 2:

We can use calculus, and evaluate the definite integral:

# A =int_a^b \ f(x) \ dx # # \ \ \ =int_0^2 \ 5x-1 \ dx # # \ \ \ =[5/2x^2-x]_0^2 # # \ \ \ =(20/2-2)-(0-0) # # \ \ \ =8 #, as before

Note:

Both of the above methods calculate the "net" area, whereas the actual area is somewhat different:

graph{(y-5x+1)(y-10000x)(y-10000x+20000)=0 [-1, 3, -5, 12]}

The actual area is:

# A = 1/2(1/5)(1) + 1/2(9/5)(9) # # \ \ \ = 1/10 + 81/10 # # \ \ \ = 82/10 # # \ \ \ = 8.2 #
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 under the curve of the function f(x) = 5x - 1 in the interval [0, 2], you need to compute the definite integral of the function over that interval.

∫[0,2] (5x - 1) dx

Now, integrate the function with respect to x:

= [5/2 * x^2 - x] evaluated from 0 to 2

Now plug in the upper and lower limits:

= [(5/2 * 2^2 - 2) - (5/2 * 0^2 - 0)]

= [(5/2 * 4 - 2) - (0)]

= [(10 - 2) - (0)]

= (8 - 0)

= 8

So, the area under the curve of f(x) = 5x - 1 in the interval [0, 2] is 8 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