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playground:playground [2021/09/30 16:34]
jojo1973 |
playground:playground [2021/10/01 09:08]
jojo1973 removed |
| where $θ_{r}\,$, $θ_{°}$ and $θ_{g}$ are the quantities entered by the user and $θ$ is what is actually fed to the trigonometric functions. | where $θ_{r}\,$, $θ_{°}$ and $θ_{g}$ are the quantities entered by the user and $θ$ is what is actually fed to the trigonometric functions. |
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| From this, two important consequences derive: the first one is that the antiderivative changes according to the angular mode. Let's consider e.g. the function $ \sin{θ} $ and apply the transformations above: | From this, two important consequences derive: |
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| * in ''[[manual:chapter6:flags:cmd_rad|RAD]]'' mode $ \sin{θ_r}\,d{θ_r} $ becomes $ \sin{θ}\,d{θ} $ whose antiderivative is $ -\cos{θ_r}\, $; | - the antiderivative changes according to the angular mode. Let's consider e.g. the function $ f(θ)=\sin{θ} $ and apply the transformations above:\\ \\ |
| * in ''[[manual:chapter6:flags:cmd_deg|DEG]]'' mode $ \sin{\left(\frac{\pi}{180}{θ_°}\right)}\,d{θ_°} $ becomes $ \sin\left(\frac{\pi}{180}{θ}\right)\frac{180}{\pi}\,d{θ}\, $ whose antiderivative is $ -\frac{180}{\pi}\cos\left(\frac{\pi}{180}{θ}\right)\, $; | * in ''[[manual:chapter6:flags:cmd_rad|RAD]]'' mode $ f(θ_r) $ is equivalent to $ f(θ) $ which differentiates to $ f(θ_r)\,dθ_r=\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=-\cos{θ}\, $; |
| * in ''[[manual:chapter6:flags:cmd_grad|GRAD]]'' mode $ \sin{\left(\frac{\pi}{200}{θ_g}\right)}\,d{θ_g} $ becomes $ \sin\left(\frac{\pi}{200}{θ}\right)\frac{200}{\pi}\,d{θ}\, $ whose antiderivative is $ -\frac{200}{\pi}\cos\left(\frac{\pi}{200}{θ}\right)\, $. | * in ''[[manual:chapter6:flags:cmd_deg|DEG]]'' mode $ f(θ_°) $ is actually $ f\left(\frac{\pi}{180}θ_°\right) $ which differentiates to $ f\left(\frac{\pi}{180}θ_°\right)\,dθ_°=\frac{180}{\pi}\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=-\frac{180}{\pi}\cos{θ}\, $; |
| | * in ''[[manual:chapter6:flags:cmd_grad|GRAD]]'' mode $ f(θ_g) $ is actually $ f\left(\frac{\pi}{200}θ_g\right) $ which differentiates to $ f\left(\frac{\pi}{200}θ_g\right)\,dθ_g=\frac{200}{\pi}\sin{θ}\,d{θ} $ and whose antiderivative is $ F(θ)=-\frac{200}{\pi}\cos{θ}\, $;\\ \\ |
| The second consequence is that no variable substitution is applied to the integration limits: of course the user can alter these limits manually, but it's not always possible or desirable. | - no inverse transformation is applied to the resulting output: this is mathematically correct, but can be disconcerting if one is not immediately aware of the implicit variable substitution. |
| | \\ |
| In conclusion, unless the user knows exactly what he/she is doing **it's advisable to perform numeric integration of trigonometric expressions in ''[[manual:chapter6:flags:cmd_rad|RAD]]'' mode**. | In conclusion, unless the user knows exactly what he/she is doing it's advisable **to perform numeric integration of trigonometric expressions in ''[[manual:chapter6:flags:cmd_rad|RAD]]'' mode**. |
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| ---- | ---- |
| ==== Example 1: Bound function on closed interval ==== | ==== Example 1: Bound function on closed interval ==== |
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| | pic | $$ \int^{2}_{0} x^{10}e^{\left(4x^{3}-3x^{4}\right)} \,dx $$ | <code>16 SETPREC | | pic | $$ \int^{2}_{0} x^{10}e^{\left(4x^{3}-3x^{4}\right)} \,dx $$ | **Input:**\\ <code>16 SETPREC |
| 'F(X)=X^10*EXP(4*X^3-3*X^4)' | 'F(X)=X^10*EXP(4*X^3-3*X^4)' |
| 0 | 0 |