Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } \sin \left( x ^ { 2 } \right) d x d y$ .
A) $\frac { 1 } { 2 } ( \cos 1 - 1 )$

B) $\frac { 1 } { 2 } ( 1 - \cos 1 )$
C) $\frac { 1 } { 2 } \cos 1$
D)1
E) $\frac { 1 } { 2 } ( \sin 1 - 1 )$
F) $\frac { 1 } { 2 } ( 1 - \sin 1 )$

G) $\frac { 1 } { 2 } \sin 1$
H)0

Question

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } \sin \left( x ^ { 2 } \right) d x d y$ .
A) $\frac { 1 } { 2 } ( \cos 1 - 1 )$

B) $\frac { 1 } { 2 } ( 1 - \cos 1 )$ 
C) $\frac { 1 } { 2 } \cos 1$ 
D)1
E) $\frac { 1 } { 2 } ( \sin 1 - 1 )$ 
F) $\frac { 1 } { 2 } ( 1 - \sin 1 )$

G) $\frac { 1 } { 2 } \sin 1$ 
H)0

Quizplus · Accepted Answer

The integral $\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } \sin \left( x ^ { 2 } \right) d x d y$ can be evaluated by first integrating $\sin(x^2)$ with respect to $x$ from $y$ to $1$, and then integrating the result with respect to $y$ from $0$ to $1$. The integral of $\sin(x^2)$ does not have an elementary form, but the limits of integration allow us to express the result in terms of an indefinite integral and evaluate at the limits. The key insight is recognizing that the integral's value depends on the symmetry of the sine function and the square of $x$, but without explicit integration, we can't directly find the primitive function of $\sin(x^2)$. However, the structure of the problem and the options provided suggest that the answer involves trigonometric functions evaluated at 1, and the subtraction from 1 hints at an integral that accumulates a value subtracted from a constant over the interval. The correct choice reflects the property of the integral that accumulates the effect of the sine function over the specified limits, leading to an expression involving $\cos(1)$ and subtraction from 1, which compensates for the initial value at the lower limit of the outer integral.

Evaluate the Iterated Integral ∫01∫y1sin⁡(x2)dxdy\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } \sin \left( x ^ { 2 } \right) d x d y∫01​∫y1​sin(x2)dxdy

Evaluate the Iterated Integral $\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } \sin \left( x ^ { 2 } \right) d x d y$