Mixing
Integration by Parts Within Multiple Integral
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In the innermost iteration of a triple integral problem, I've applied two instances of integration by parts, and ended up with$$int_0^{pi/6}int_0^{pi}xint_0^1ysin(yz) dy dx dz = int_0^{pi/6}int_0^{pi}-frac{x}{z^2}(int_0^1ysin(yz) dy + int_0^1frac{cos(yz)}{z} - sin(yz) dy) dx dz$$In single variable integration, the next step would be to combine the original integral with the like term produced by the double integration by parts. However, in this case I have the encapsulating integrals to worry about. Furthermore, I'm not sure how careful I have to be with the constants $x$ and $z$ while manipulating things, since later they won't be constants anymore. Is there a clear way to proceed, or is this a sign that I've made a mistake or chosen the wrong iteration order?
integration multivariable-calculus multiple-integral iterated-integrals
asked Jan 14 at 3:37
user10478user10478
453211
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add a comment |
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In the innermost iteration of a triple integral problem, I've applied two instances of integration by parts, and ended up with$$int_0^{pi/6}int_0^{pi}xint_0^1ysin(yz) dy dx dz = int_0^{pi/6}int_0^{pi}-frac{x}{z^2}(int_0^1ysin(yz) dy + int_0^1frac{cos(yz)}{z} - sin(yz) dy) dx dz$$In single variable integration, the next step would be to combine the original integral with the like term produced by the double integration by parts. However, in this case I have the encapsulating integrals to worry about. Furthermore, I'm not sure how careful I have to be with the constants $x$ and $z$ while manipulating things, since later they won't be constants anymore. Is there a clear way to proceed, or is this a sign that I've made a mistake or chosen the wrong iteration order?
integration multivariable-calculus multiple-integral iterated-integrals
asked Jan 14 at 3:37
user10478user10478
453211
$endgroup$
add a comment |
$begingroup$
In the innermost iteration of a triple integral problem, I've applied two instances of integration by parts, and ended up with$$int_0^{pi/6}int_0^{pi}xint_0^1ysin(yz) dy dx dz = int_0^{pi/6}int_0^{pi}-frac{x}{z^2}(int_0^1ysin(yz) dy + int_0^1frac{cos(yz)}{z} - sin(yz) dy) dx dz$$In single variable integration, the next step would be to combine the original integral with the like term produced by the double integration by parts. However, in this case I have the encapsulating integrals to worry about. Furthermore, I'm not sure how careful I have to be with the constants $x$ and $z$ while manipulating things, since later they won't be constants anymore. Is there a clear way to proceed, or is this a sign that I've made a mistake or chosen the wrong iteration order?
integration multivariable-calculus multiple-integral iterated-integrals
asked Jan 14 at 3:37
user10478user10478
453211
$endgroup$
In the innermost iteration of a triple integral problem, I've applied two instances of integration by parts, and ended up with$$int_0^{pi/6}int_0^{pi}xint_0^1ysin(yz) dy dx dz = int_0^{pi/6}int_0^{pi}-frac{x}{z^2}(int_0^1ysin(yz) dy + int_0^1frac{cos(yz)}{z} - sin(yz) dy) dx dz$$In single variable integration, the next step would be to combine the original integral with the like term produced by the double integration by parts. However, in this case I have the encapsulating integrals to worry about. Furthermore, I'm not sure how careful I have to be with the constants $x$ and $z$ while manipulating things, since later they won't be constants anymore. Is there a clear way to proceed, or is this a sign that I've made a mistake or chosen the wrong iteration order?
integration multivariable-calculus multiple-integral iterated-integrals
integration multivariable-calculus multiple-integral iterated-integrals
asked Jan 14 at 3:37
user10478user10478
453211
asked Jan 14 at 3:37
user10478user10478
453211
edited Jan 16 at 3:29
user10478
asked Jan 14 at 3:37
user10478user10478
453211
asked Jan 14 at 3:37
user10478user10478
453211
asked Jan 14 at 3:37
user10478user10478
453211
453211
add a comment |
add a comment |
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