Find the area between two curves

Two curves cross inside the interval. What should be done before integrating?

Move k through −1, 0 and 1. The dashed graph qₖ(x)=x²+kx always meets F(x)=x² at O=(0,0). Compare their directions there. Which k makes their slopes agree as well as their heights?

For which k do both graphs share their tangent at O?

Area between two curves

The area between two curves is found from vertical widths. Find all intersections, determine which curve is above on each piece, integrate upper minus lower, and add the positive contributions

1. 1Find the intersections Set the two function values equal to locate every crossing \(\displaystyle 6x=3x^2\Rightarrow x=0,2\)
2. 2Split at every crossing Use a separate interval whenever the upper curve changes \(\displaystyle [-1,3]=[-1,0]\cup[0,2]\cup[2,3]\)
3. 3Compare curve heights Use a test input in each piece to identify the upper curve \(\displaystyle f(1)=6>g(1)=3\)
4. 4Integrate positive widths Subtract the lower curve from the upper curve on each piece and add the areas \(\displaystyle A=\int_0^2\left(6x-3x^2\right)\,dx=4\)

Curves and interval

\(\displaystyle f(x)=6x,\quad g(x)=3x^2,\quad -1\le x\le3\)

Find the second intersection x-coordinate of the two curves

Curves and interval

\(\displaystyle f(x)=6x,\quad g(x)=3x^2,\quad -1\le x\le3\)

Calculate the area contribution on the central piece from x=0 to x=2.

Ignoring intersections. One of these lines is right. Which one?

Translate equal height and slope into separate equations

To make two differentiable graphs share their tangent at input u, require both the same function value and the same derivative value there. Two distinct contact inputs u≠v give four equations for a polynomial of degree at most three. Do not assume the leading coefficient is nonzero: valid data can force a quadratic or a constant. Matching slopes without heights can give parallel, distinct tangent lines

1. 1Find the point At input t, the point on the curve has coordinates shown below \(\displaystyle P=\left(t,f(t)\right)\)
2. 2Find the tangent gradient Differentiate the function and substitute the x-coordinate of the point \(\displaystyle m_{\mathrm{t}}=f'(t)\)
3. 3Use point-gradient form Substitute the point and the required gradient into the equation of a straight line \(\displaystyle y-y_0=m\left(x-x_0\right)\)

Worked fit: four conditions and four checks

Fit p to F(x)=eˣ at inputs 0 and 1, matching both values and slopes. At zero, d=1 and c=1. At one, a+b=e−2 and 3a+2b=e−1. Subtracting twice the first equation from the second gives a=3−e, then b=2e−5. Check p(0)=1, p′(0)=1, p(1)=e and p′(1)=e. This is a genuine cubic because 3−e≠0. These four conditions do not say that the functions are equal at other inputs

1. 1Keep the complete response Write the coefficient equations and solve them, check every height and slope condition, justify curve order, set up and evaluate the area, and record your radian-mode numerical work with independent signs. The complete derivation, graph, technology record and completeness proof are ungraded. Short checked values assess only components

Your complete circular fit and enclosed-area response

Let F(x)=cos x. Fit p(x)=ax³+bx²+cx+d to F at both −π/2 and π/2, matching function values and derivatives. Write and solve all four equations, check both contact heights and slopes, and state the resulting degree. On the closed interval [−π/2,π/2], sketch both graphs with their common endpoints and explain which is above throughout, not just at one sample input. Hence set up and evaluate the enclosed area exactly, then report it to three decimal places

Conditions or certified enclosure

\(\displaystyle p(x)=ax^3+bx^2+cx+d;\quad p(\pm\pi/2)=0,\quad p^{\prime}(-\pi/2)=1,\quad p^{\prime}(\pi/2)=-1\)

After solving the four equations, what is the coefficient of x³?

Review all four coefficient equations

Write h=π/2. The two height equations are ah³+bh²+ch+d=0 and −ah³+bh²−ch+d=0. They imply ah²+c=0 and bh²+d=0. The slope equations are 3ah²+2bh+c=−1 and 3ah²−2bh+c=1. Adding gives 3ah²+c=0; subtracting gives 4bh=−2. Hence a=c=0, b=−1/π and d=π/4. Directly, p(±π/2)=0 and p′(−π/2)=1, p′(π/2)=−1, as required

1. 1Keep the complete response Write the coefficient equations and solve them, check every height and slope condition, justify curve order, set up and evaluate the area, and record your radian-mode numerical work with independent signs. The complete derivation, graph, technology record and completeness proof are ungraded. Short checked values assess only components

Fitted parabola and interval

\(\displaystyle p(x)=\frac{\pi}{4}-\frac{x^2}{\pi},\qquad -\frac{\pi}{2}\le x\le\frac{\pi}{2}\)

The graph suggests that cosine is above the fitted parabola. Which argument establishes this throughout the interval?

Prove the order, then integrate the positive difference

Let D=cos x−π/4+x²/π. It is even. On [0,π/2], sine is concave because its second derivative is −sin x≤0, so sin x lies above its chord 2x/π. Hence D′≤0 and D(π/2)=0 imply D≥0. The inequalities are strict inside the interval, so the endpoints are the only intersections there. Integrating cosine minus the parabola therefore gives A=2−π²/6≈0.355065933152, or 0.355 to three decimal places. A signed integral without the curve-order argument would not by itself establish the enclosed area

1. 1Keep the complete response Write the coefficient equations and solve them, check every height and slope condition, justify curve order, set up and evaluate the area, and record your radian-mode numerical work with independent signs. The complete derivation, graph, technology record and completeness proof are ungraded. Short checked values assess only components

Conditions or certified enclosure

\(\displaystyle A=2-\pi^2/6\)

After completing the fit and area proof, report the enclosed area to exactly three decimal places

Continue: find every further common tangent contact

Keep F(x)=cos x, now considered on all real inputs. A new family q_b(x)=1+bx² already matches its height and slope at zero. Find every pair (b,r) for which it also matches both at a further input r∈(0,3π). Derive one equation in r by eliminating b, factor it, and retain any roots lost by division. Prove how many admissible contacts there are on the whole interval. Keep exact roots exact; for a non-exact root give a three-place value, an independent sign bracket and b to six decimal places. Substitute into both original conditions and explain why the fitted constant case is allowed. Keep your full equations and radian-mode numerical record

Conditions or certified enclosure

\(\displaystyle 1+br^2=\cos r,\quad2br=-\sin r,\quad0<r<3\pi\)

After your completeness proof, how many distinct admissible pairs (b,r) satisfy both contact conditions?

Eliminate without discarding the exact contact

Since r>0, the slope condition gives b=−sin r/(2r). Substitution into the height condition gives r sin r−2(1−cos r)=0. Factoring with half-angle identities gives two branches. The branch sin(r/2)=0 yields r=2π and b=0; q is then the constant 1, which matches cosine in both height and slope there. On the other branch, cos(r/2) cannot be zero: substituting r=π into the unfactored residual gives −4, and 3π is excluded. Thus division gives tan(r/2)=r/2. Never divide by sin(r/2) before retaining its zero branch

Prove the complete list and certify the numerical contact

Set u=r/2. On (0,π/2), tan u−u is strictly positive because its derivative is tan²u and its limit at zero is zero. On (π/2,π), tan u is negative while u is positive. On (π,3π/2), tan u−u increases strictly from −π to infinity, so exactly one zero occurs. The half-angle identity and tan(r/2)=r/2 give sin r=4r/(r²+4). Thus r≈8.986818915818 and b=−2/(r²+4)≈−0.023595224613, hence b=−0.023595 to six places. Independent signs for K(r)=r sin r−2(1−cos r) are K(8.98681)>0 and K(8.98682)<0. Both bracket endpoints round to 8.987; the corresponding b enclosure lies wholly inside the six-place rounding cell for −0.023595. Together with the exact pair (0,2π), these are all solutions

Conditions or certified enclosure

\(\displaystyle 8.98681<r<8.98682\)

Use the independently certified enclosure to report the non-exact contact input to exactly three decimal places