Urgent! Calculus Help Please!

Question

1. Evaluate the integral S_-1² f(x)dx, when f(x)=3-x³ when x is less than or equal to 1; f(x)=x² when x is greater than one.

 

 

2. Evaluate the integrals using the FTC 1.

(a) S_-2^-1 (1/x⁵)dx

(b) S₁⁴ (5/t²)dt

Answer 1

Integral means to find the anti-derivative.  Anti-derivative means to find a function, in which when find its derivative, you get the function you are integrating. 

 

Use this rule to evaluate definite integrals:

 

∫_a ^b f(x)dx = F(b) - F(a)

 

where F(x) is the anti-derivative of f(x).  This means if we take the derivative of F(x), we get f(x).

 

a is the lower bound and b is the upper bound.

 

 

1)

 

You are given a general integral with general bounds to evaluate using specific functions with specific bounds.

 

So when f(x)=3-x³ , you will evaluate     ∫_-1 ¹ (3 - x³) dx

 

When f(x)=x² , you will evaluate    ∫₁ ² (x²) dx

 

 

For the first integral:

 

F(x) = -(1/4)x⁴

 

Now evaluate        F(1) - F(-1)

 

 

For the second integral:

 

F(x) = (1/3)x³

 

Now evaluate        F(2) - F(1)

 

 

2a)

 

∫₂ ¹ (x^-5)dx

 

 

Use the rule to evaluate the definite integral.

 

 

2b)

 

 

When we have a constant in the integral, we take out the constant and put it in front of the integral sign.  So you are multiplying the constant by the integral.

 

5∫₁ ⁴ (t^-2) dt

 

Use the rule to evaluate the definite integral.