Let be a positive real constant

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

k

be a positive real constant. Two curves are defined by

L_k: x=ky

and

P: x=y^{2}+1

. For

k>0

the line

L_k

and the parabola

P

intersect at two distinct

y

–values, enclosing a finite region

T(k)

between them. (a) Show that the

y

–coordinates of the intersection points of

L_k

and

P

are the real roots of the quadratic equation

y^{2}-ky+1=0

. Express these roots

y_1<y_2

explicitly in terms of

k

. (b) Assuming

k>0

so that

L_k

forms the right‐hand boundary of

T(k)

between

y_1

and

y_2

, set up—but do not evaluate—an integral expression for the area

A(k)

of region

T(k)

in terms of

k

. Your integrand should be written in simplest algebraic form. (c) Evaluate the integral from part (b) to obtain a simplified closed‐form expression for

A(k)

as a function of

k

. (d) Determine all positive values of

k

for which

A(k)=10

. Provide exact values or justify why none exist.

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