find the definite integral of sqrt(9 + x^2) between points 3 and -3

[hndalama]

The book is asking us to solve it by using a table to find the antiderivative and then solve for the value of the definite integral.
The answer is represented as 3 sqrt(18) +9/2 ln[(3 +sqrt(18) / (sqrt(18) -3)] when I plug this into the calculator I get the answer 20.66. This is derived using the antiderivative pattern: integral of (x²+ a²) = x/2 sqrt(x² + a²) + a²/2 ln[x + sqrt(x² + a²) ] + C

I recognise the integrand, 9 + x² to be the equation of a semi circle with radius 3. I understand finding the definite integral between points 3 and -3 to be equivalent to finding the area of the semicircle. But when I calculate this area, 9pi /2 = 14.1. Why are these answers different?

 

[Deleted member 4993]

The book is asking us to solve it by using a table to find the antiderivative and then solve for the value of the definite integral.
The answer is represented as 3 sqrt(18) +9/2 ln[(3 +sqrt(18) / (sqrt(18) -3)] when I plug this into the calculator I get the answer 20.66. This is derived using the antiderivative pattern: integral of (x²+ a²) = x/2 sqrt(x² + a²) + a²/2 ln[x + sqrt(x² + a²) ] + C

I recognise the integrand, 9 + x² to be the equation of a semi circle with radius 3. I understand finding the definite integral between points 3 and -3 to be equivalent to finding the area of the semicircle. But when I calculate this area, 9pi /2 = 14.1. Why are these answers different?

You have:

y=√(x²+3²)

y² = x² + 3²

y² - x² = 3²

That is NOT an equation of a circle!

 

[Steven G]

The book is asking us to solve it by using a table to find the antiderivative and then solve for the value of the definite integral.
The answer is represented as 3 sqrt(18) +9/2 ln[(3 +sqrt(18) / (sqrt(18) -3)] when I plug this into the calculator I get the answer 20.66. This is derived using the antiderivative pattern: integral of (x²+ a²) = x/2 sqrt(x² + a²) + a²/2 ln[x + sqrt(x² + a²) ] + C

I recognise the integrand, 9 + x² to be the equation of a semi circle with radius 3. I understand finding the definite integral between points 3 and -3 to be equivalent to finding the area of the semicircle. But when I calculate this area, 9pi /2 = 14.1. Why are these answers different?

Are you saying that 9 + 3² = 0 and 9 + (-3)² = 0?

$\displaystyle x^2 + y^2 = r^2 => y^2=r^2-x^2 => y= \pm \sqrt{r^2-x^2}$

 

[hndalama]

oh thanks guys