Question

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

Accepted Answer

Recall that the Fresnel integrals are defined as $$S(x) = \$$int_0$$^x \sin(t^2$) \, dt$$ and $$C(x) = \$$int_0$$^x \cos(t^2$) \, dt. To $$approximate these integrals, we use polynomial approximations (Taylor series) of the integrands $$\sin(t^2$)$$ and $$\cos(t^2$)$$ from part (b). Write down the Taylor series expansions for $$\sin(t^2$)$$ and $$\cos(t^2$)$$ around $$t=0. $$For example, $$\sin(t^2$)$$ can be expanded as $$t^2$$ - \frac{$$t^6$$}{3!} + \frac{$$t^{10}$$}{5!} - \cdots$$, and $$\$$cos(t^2$$)$$ as 1$$ - \frac{$$t^4$$}{2!} + \frac{$$t^8$$}{4!} - \cdots$$. Substitute these polynomial approximations into the integrals for S(x$$)$$ and C(x$$)$$, so that S(x$$) \approx \$$int_0^x$$ \left( $$t^2$$ - \frac{$$t^6$$}{3!} + \frac{$$t^{10}$$}{5!} - \cdots \right) dt$$ and C(x$$) \approx \$$int_0^x$$ \left( 1 - \frac{$$t^4$$}{2!} + \frac{$$t^8$$}{4!} - \cdots \right) dt$$. Integrate each term of the polynomial inside the integral with respect to t$ from 0 to x$. For example, $$\$$int_0^x$$ $$t^2 dt$$ = \frac{$$x^3$$}{3}$$, $$\$$int_0^x$$ $$t^6 dt$$ = \frac{$$x^7$$}{7}$$, and so on. This will give you polynomial expressions in terms of x$. Finally, plug in the values x=0.05$ for S(0.05$$)$$ and x=-0.25$ for C(-0.25$$)$$ into the resulting polynomial expressions to get the approximate values of the Fresnel integrals.

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

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{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integralsS(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dtc. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

Verified video answer for a similar problem:

Key Concepts

Fresnel Integrals

Polynomial Approximation of Functions

Evaluating Definite Integrals Using Approximations

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).