Evaluate the line integral, where c is the given curve. c y3 ds,
c. x = t3, y = t, 0 ≤ t ≤ 4

∫C) y³ × ds = 7962624,02 surface units

∫C) y³ × ds .               where  C    x = t³     y = t      with    0 ≤ t ≤ 4

∫C) y³ × ds   = ∫∫R) F(r(t) × dr

Then  F (r(t)) =   y³ . = t³ .          r ( t³ , t )  then .    dr/dt = (3×t² , 1 )

|| dr/dt|| = √ ( 3×t²)² + (1)²

|| dr/dt|| = √9×t⁴ + 1

∫C) y³ × ds   = ∫∫R) F(r(t) × dr . =  ∫₀⁴ t³ ×√9×t⁴ + 1    ×dt

To solve the integral we make a change of variables, we call

9×t⁴ + 1  = v .  then .    36×t³ ×dt = dv

Now as we change variables we need to change the limits of integration as follows:If   9×t⁴ + 1 = v .       then when t = 0   v = 1

And when  t = 4 .                then . v = 9×(4)⁴ + 1 . = 2305

The integral takes that form

∫₁²³⁰⁵ [ 1/36)× √v × dv . =   (1/36) ×∫₁²³⁰⁵ (v)¹/² dv  

= (1/36)×(2/3) × v³/² |₁²³⁰⁵ = (2/108)× √v³|₁²³⁰⁵ =

= (2/108) [ √( 9×t⁴ + 1 )³|₀⁴ =  (2/108) [ √( 9⁴×4¹⁶ + 1  - √(9)⁴×0 + 1)

=  (2/108) [√( (9⁴×4¹⁶ + 1 ) . + 1 ]

Neglecting  in √( (9⁴×4¹⁶ + 1 )  1 unit .  we get 9²×4⁸ = 429981696

= (2/108) [429981697]

= 7962624,02

i used to know how to do this in 7th but i forget easily

step-by-step explanation:

you forgot the options, but i found them anyways, the correct answer is "john wilson believes that words and actions based on the intellect are superior to those based on emotions".

john wilson is depicted as an extremely intelligent man. the point of the excerpt is to make a sort of distinction between emotional and rational intelligence. he happens to have both attributes, but he clearly favors his intellectual gifts. at the same time, because of his religious background, he's a little bit conflicted with this and it makes him ashamed.

hope this !