Exploring Arc Length in Definite Integrals

What is the function whose arc length we are calculating?

• y = 3x/2

• y = x^2

• y = x^(2/3)

• y = x^(3/2)

What is the upper limit of x for calculating the arc length?

• 32/9

• 9/32

• 3.5

• 9/3

Why was the number 32/9 specifically chosen for x?

• Because it is the function's maximum

• To demonstrate a complex example

• Random selection

• To simplify the calculations

What formula is used to calculate the arc length of a curve?

• The derivative of the function squared

• The square root of 1 plus the derivative of the function squared

• The integral of the function from 0 to x

• x times the derivative of the function

What is the derivative of the function y = x^(3/2)?

• 3/2 x^(1/2)

• 3x^(2/3)

• 2/3 x^(1/2)

• x^(3/2)

What substitution is made for u-substitution?

• u = x^(3/2)

• u = 1 + 9/4 x

• u = 3/2 x^(1/2)

• u = 9/4 x

What are the new bounds for u after substitution?

• 0 to 8

• 1 to 8

• 0 to 32/9

• 1 to 9

What is the purpose of u-substitution in this problem?

• To simplify the integral for easier calculation

• To determine the function's maximum value

• To calculate the bounds of integration

• To find the derivative of the function

What is the antiderivative of the square root of u?

• 3/2 u^(1/2)

• 2/3 u^(1/2)

• u^(3/2) divided by (3/2)

• u^(2/3)

What is the final value of the arc length?

• 26/27

• 27/26

• 27/208

• 208/27