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Question Number 203881 by mr W last updated on 31/Jan/24

Commented by mr W last updated on 31/Jan/24

$$findthehatchedareabetweencurve$$$$anditstangentline.$$

Answered by Frix last updated on 01/Feb/24

$$\mathrm{Curve}:f\left(x\right)=x^4+{ax}^3+{bx}^2+cx+d$$$$\mathrm{Tangent}:g\left(x\right)=px+q$$$$f\left(3\right)=g\left(3\right)$$$$f\left(9\right)=g\left(9\right)$$$$f^{\prime }\left(3\right)=g^{\prime }\left(3\right)$$$$f^{\prime }\left(9\right)=g^{\prime }\left(9\right)$$$$\Rightarrow$$$$f\left(x\right)=x^4-24x^3+198x^2+cx+d$$$$g\left(x\right)=648x-729+cx+d$$$$\underset{3}{\stackrel{9}{\int }}f\left(x\right)-g\left(x\right)dx=\frac{1296}{5}$$

Commented by mr W last updated on 01/Feb/24

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Answered by mr W last updated on 01/Feb/24

Commented by mr W last updated on 01/Feb/24

$$curve:f\left(x\right)=x^4+{ax}^3+{bx}^2+cx+d$$$$tangent:g\left(x\right)=hx+k$$$$f\left(x\right)-g\left(x\right)isalsoaquatricequation$$$$whichhastwodoublerealroots:$$$$x=p=3,x=q=9$$$$\Rightarrow f\left(x\right)-g\left(x\right)={(x-p)}^2{(x-q)}^2$$$$A={\int }_p^q[f\left(x\right)-g\left(x\right)]dx$$$$={\int }_p^q{(x-p)}^2{(x-q)}^{2}dx$$$$={\int }_0^{q-p}{u}^2{(u-(q-p))}^{2}duwithu=x-q$$$$={\int }_0^s{u}^2{(u-s)}^{2}duwiths=q-p$$$$={\int }_0^s(u^4-2{su}^3+s^2{u}^2)du$$$$=\frac{s^5}{5}-\frac{s^5}{2}+\frac{s^5}{3}$$$$=\frac{s^5}{30}$$$$withp=3,q=9$$$$\Rightarrow s=9-3=6$$$$\Rightarrow A=\frac{6^5}{30}=\frac{1296}{5}✓$$

Commented by Frix last updated on 01/Feb/24

$$\mathrm{Nice}!$$

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