Calculus problem #2053

Calculus Level 2

Let C C be the arc of the curve y = x y=\sqrt{x} from points ( 1 , 1 ) (1, 1) to ( 4 , 2 ) (4, 2) . What is the value of N = ∫ C ( x 2 y 3 − x ) d y ? N = \int_C (x^2y^3-\sqrt{x})\ dy?


The answer is 30.375.

View wiki
Gene Keun Chung by Gene Keun Chung , 50, South Korea

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Tunk-Fey Ariawan
Jan 31, 2014

We have x = y 2 \,x=y^2 for 1 ≤ y ≤ 2 \,1\le y\le 2 . Hence, the solution is N = ∫ 1 2 ( y 7 − y ) d y = 1 8 y 8 − 1 2 y 2 ∣ 1 2 = 1 8 ( 2 8 − 1 ) − 1 2 ( 2 2 − 1 ) = 30.375 \begin{aligned} N &= \int_{1}^{2} (y^7-y)\,dy\\ &= \left.\frac{1}{8} y^8 - \frac{1}{2} y^2\right|_1^2\\ &= \frac{1}{8}(2^8-1)-\frac{1}{2}(2^2-1)\\ &=\boxed{30.375} \end{aligned} # Q . E . D . # \text{\# }\mathbb{Q}.\mathbb{E}.\mathbb{D}.\text{\#}

3 Helpful 0 Interesting 1 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...