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We assume ${\\lVert f_{xy}\\rVert}_p$ is bounded (integration over $[a,b]\\times[c,d]$), assume ${\\lVert f_x(\\cdot,c)\\rVert}_p$ and ${\\lVert f_x(\\cdot,d)\\rVert}_p$ are bounded (integration over $[a,b]$), and assume ${\\lVert f_y(a,\\cdot)\\rVert}_p$ and ${\\lVert f_y(b,\\cdot)\\rVert}_p$ are bounded (integration over $[c,d]$). The methods are elementary, using only integration by parts and H\\\"older's inequal"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1905.05805","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-05-14T19:25:08Z","cross_cats_sorted":[],"title_canon_sha256":"2b8ba1093d810338a384359e0439209c6d55755f5c6056a9ad1d1ae5e1ab8487","abstract_canon_sha256":"c44262094ead69448cc64bd0895e8bb2e6c9f3445fb266a891eb1340064baff0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:08.491327Z","signature_b64":"QiXRrSzHnV6QNtzY94muR2l91EcFBPlIUdIJuHXW3IURpuydyRykVxxexv84WQZCb9IRf4eUQZ3XyAjs7ybADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ecdc9fd0a7b61ad602071a448738f1c05db2ab80b69ad9f654a2b38550f39dbe","last_reissued_at":"2026-05-17T23:46:08.490809Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:08.490809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Elementary numerical methods for double integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Cameron Grant, Erik Talvila","submitted_at":"2019-05-14T19:25:08Z","abstract_excerpt":"Approximations to the integral $\\int_a^b\\int_c^d f(x,y)\\,dy\\,dx$ are obtained under the assumption that the partial derivatives of the integrand are in an $L^p$ space, for some $1\\leq p\\leq\\infty$. We assume ${\\lVert f_{xy}\\rVert}_p$ is bounded (integration over $[a,b]\\times[c,d]$), assume ${\\lVert f_x(\\cdot,c)\\rVert}_p$ and ${\\lVert f_x(\\cdot,d)\\rVert}_p$ are bounded (integration over $[a,b]$), and assume ${\\lVert f_y(a,\\cdot)\\rVert}_p$ and ${\\lVert f_y(b,\\cdot)\\rVert}_p$ are bounded (integration over $[c,d]$). 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