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We studied the random variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mi>t</mi></msub><mo>=</mo><msub><mi>vol</mi><msup><mi>S</mi><mn>2</mn></msup></msub><mrow><mo>(</mo><msub><mi>g</mi><mi>t</mi></msub><mi>B</mi><mo>∩</mo><mi>B</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>B</i> is a disc on the sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mn>2</mn></msup></semantics></math></inline-formula> centered at the north pole and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>g</mi><mi>t</mi></msub><mo>)</mo></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> is the Brownian motion on the special orthogonal group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>t</mi></msub></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> is the first time when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>V</mi><mi>t</mi></msub></semantics></math></inline-formula> vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Γ</mi><mi>B</mi></msub></semantics></math></inline-formula> seen as the support of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>vol</mi><msup><mi>S</mi><mn>2</mn></msup></msub><mrow><mo>(</mo><mi>g</mi><mi>B</mi><mo>∩</mo><mi>B</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> immersed in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The integral formula depends on the mean curvature of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Γ</mi><mi>B</mi></msub></semantics></math></inline-formula> and the diameter of <i>B</i>.