Could someone please solve this maths problem and provide some workings?

Question

Find the unsigned area between the points x = -1 and x = 3 for the curve y = 2 - 2e^(-x)

Answer 1

The area under a curve can be found with the integral of that curve, from one point to another, so from -1 to 3:
integral from -1 to 3 of (2 - 2 e^(-x) ) dx

The integral of (2 - 2*e^(-x) dx) is (2*e^(-x) + 2x), you can find this by knowing that the derivative of (1/e^x) = -1 * (1/e^x), and then just doing the opposite of the power rule with 2, finding 2x, thus the integral of (2 - 2e^(-x)) = (2e^(-x) + 2x). Then you just plug in the numbers, plug in the upper limit into the first one, and subtract the lower limit plugged into the second one.

(2 * e^(-(3)) + 2(3)) - (2 * e^(-(-1)) + 2(-1))
2 * e^(-3) + 6 - 2e + 2
2 * e^(-3) - 2e + 8
2(e^-3 - e) + 8

Or, an approximate answer:
2.66301

You said unsigned, so that would be the absolute value of the answer just found, but because it is positive, theres no need to worry about that.

Edit: If, unsigned means that all the ones below the x axies turn into positive before combining, then you could have to integrate by parts, because from -1 to 0, it is below the x axies, so:
abs( integral from -1 to 0 of (2 - 2 e^(-x) ) dx ) + integral from 0 to 3 of (2 - 2 e^(-x) ) dx

abs( (2*e^(-(0)) + 2(0)) - (2*e^(-(-1)) + 2(-1)) ) + ( (2*e^(-(3)) + 2(3)) - (2*e^(-(0)) + 2(0)) )

abs( (2) - ((2e) - 2) ) + ( (2*e^(-(3)) + 6) - (2) )

abs( 2 - (2e) + 2 ) + 2*e^(-3) + 6 - 2

abs( 4 - 2e) + 2*e^(-3) + 4

2e + 2e^(-3)

That last part can then be approximated to:
5.53614

Answer 2

You need to know where the curve crosses the x axis (do this by solving for x when y=0). Let x=A be where it crosses. If -1
If A is not between -1 and 3 then it's just the absolute value of the integral from -1 to 3.

Answer 3

You integrate the function y = f(x) and evaluate at x=-1 and x=3. This gives you the area "under the curve and above the y=0 line. The integral is
2(x+e^(-x))

Answer 4

integrate the equation from -1 to 0. take the absolute value of that value and add it to the equation integrate from 0 to 3.